Powers and Roots: Complete the missing numbers

Let's simplify the direct calculation of the left side of the equation:

$6^1+1^6+\sqrt{81}=\textcolor{red}{☐}^2 \\ 6+1+9=\textcolor{red}{☐}^2\\ 16=\textcolor{red}{☐}^2\\$When we calculated the numerical value of the term with the exponent and the term with the root, and remembered that raising the number 1 to any power will always give the result 1,

Now let's examine the equation that we obtained. On the left side we have the number 16 and on the right side we have a number (unknown) raised to the second power,

So we ask the question: "What number do we need to square to get the number 16?"

And the answer to that is of course - the number 4,

Therefore:

$16=\textcolor{red}{4}^2$However, since we're dealing with an even power (power of 2), we must also consider the negative possibility,

Meaning it also holds true that:

$16=\textcolor{red}{(-4)}^2$

Therefore, the correct answer is answer C.

Calculate the values of all terms in the equation using direct computation:

$7^1+3^4=4^3+\sqrt{\textcolor{red}{☐}}+2^3 \\ 7+81=64+\sqrt{\textcolor{red}{☐}}+8$

Remember that raising any number to the power of 1 will always give the number itself,

Note that this is an equation and therefore we can move terms from one side to the other, doing so whilst remembering that when a term moves sides it changes its sign:

$7+81=64+\sqrt{\textcolor{red}{☐}}+8 \\ 7+81-64-8=\sqrt{\textcolor{red}{☐}}\\ 16 =\sqrt{\textcolor{red}{☐}}$

In the final step, we simplified the left side of the equation by combining like terms,

Now let's examine the equation that we obtained:

On the left side we have the number 16 and on the right side we have a number (which is unknown) under a square root,

Therefore we ask the question: "The square root of which number is 16?"

We can answer this question by guessing and checking the square roots of different numbers using a calculator, but a better way is to remember that square root and squaring are inverse operations and therefore:

$16=\sqrt{16^2}$

Therefore the answer to the above question is of course the number:

$16^2$

Let's calculate the numerical value of this term:

$16^2=256$

Therefore the answer to the above question, meaning - the unknown number under the square root in the problem, is the number 256:

$16 =\sqrt{\textcolor{red}{256}}$

The correct answer is answer B.

Let's recall two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for exponents within parentheses where terms are multiplied, but in the opposite direction:

$x^n\cdot y^n =(x\cdot y)^n$Unlike previous questions where we converted the square root to its corresponding half power according to the definition of root as an exponent, in this problem we will not make this conversion but rather understand and internalize that according to the definition of root as an exponent, the root is an exponent in every way and therefore all laws of exponents apply to it, particularly the law of exponents mentioned in b',

We'll start then by simplifying the expression in the right parentheses (as written there, actually this is the denominator of the fraction equivalent to the division operation, we'll emphasize this later), we'll simplify this expression while applying the above understanding and using the law of exponents mentioned in b':

$(\sqrt{5}\cdot\sqrt{2})^2=(\sqrt{5\cdot2})^2=(\sqrt{10})^2$where in the first stage we applied the above understanding and treated the square root (which is actually a half power) as an exponent in every way and applied the law of exponents mentioned in b', we applied this law in the direction specified there by noting that the two multiplication terms have the same exponent (half power - of the square root) and therefore we could combine them together as multiplication between the bases under the same root (same exponent), in the next stage we simplified the expression under the root,

Now we'll remember that square root and square power are inverse operations and therefore cancel each other out, therefore:

$(\sqrt{10})^2 =10$Now we'll return to the complete expression on the right side of the equation and insert this information, in the next stage we'll write the division operation as a fraction:

$\sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{5}\cdot\sqrt{2})^2 \\ \downarrow\\ \sqrt{\textcolor{red}{☐}}=(2^5+18):10\\ \sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10}$$$Note that the division operation acts on the parentheses and therefore refers to them in their entirety, therefore the entire expression in parentheses enters the numerator of the fraction,

Let's continue and simplify the expression in the numerator of the fraction on the right side:

$\sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{32+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{\not{50}}{\not{10}} \\ \sqrt{\textcolor{red}{☐}}=5$where in the first stage we calculated the numerical value of the term with the exponent in the numerator of the fraction and in the next stage we performed the addition operation in the numerator, in the final stage we reduced the resulting fraction (performing the division operation in fact),

Let's summarize the solution steps so far, we got that:

$\sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{5}\cdot\sqrt{2})^2 \\ \sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{10})^2 \\ \sqrt{\textcolor{red}{☐}}=(2^5+18):10\\ \downarrow\\ \sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{\not{50}}{\not{10}} \\ \sqrt{\textcolor{red}{☐}}=5$

Let's examine now the expression we got:

On the right side we got the number 5, and on the left side there's a (unknown) number that's under a square root,

In other words - we need to answer the question - "The square root of which number is 5?"

The answer to this question is of course the number: 25, it's easy to verify this by calculating its square root which indeed gives the result 5,

Meaning - for the equation to be true, under the root must be the number 25:

$\sqrt{\textcolor{red}{25}}=5$Therefore the correct answer is answer d.

To solve the problem, we need to evaluate both sides of the equation step by step and determine which number should replace the missing value to maintain equality. We'll start by evaluating the left side and simplifying the right side to find the missing number.

Let's examine the left side of the equation:
$(2^3+5^2)-9^2:3^2-\sqrt{100}$

• First, calculate $2^3$ which is $8$.

• Next, calculate $5^2$ which is $25$.

• Add these results together: $8 + 25 = 33$.

• Calculate $9^2$ which is $81$.

• Calculate $3^2$ which is $9$.

• Divide $81$ by $9$:$\ 81 \div 9 = 9$.

• Subtract $9$ from $33$: $33 - 9 = 24$.

• Calculate $\sqrt{100}$ which is $10$.

• Subtract $10$ from $24$:$24 - 10 = 14$.

Now, let's simplify the right side of the equation:
$(6\cdot5-\textcolor{red}{☐}^2)^2-\sqrt{25}-6$

• Calculate $6\cdot5$ which is $30$.

• The expression becomes: $(30-\textcolor{red}{☐}^2)^2-\sqrt{25}-6$.

• We know from the original problem statement that the complete expression must equal $14$.

• Calculate $\sqrt{25}$, which is $5$.

• The equation: $(30-\textcolor{red}{☐}^2)^2-5-6=14$.

• Simplify further:
  $((30-\textcolor{red}{☐}^2)^2=25$

• Taking square roots of both sides: $30-\textcolor{red}{☐}^2=5$ or $-(30-\textcolor{red}{☐}^2)=5$

• Solving for $\textcolor{red}{☐}$:

  • $30-\textcolor{red}{☐}^2=5\rightarrow \textcolor{red}{☐}^2=25\rightarrow \textcolor{red}{☐}=5$

  • Alternatively, for the negative case: $-(30-\textcolor{red}{☐}^2)=5\rightarrow 30-\textcolor{red}{☐}^2=-5$ does not result in a real number solution.

Therefore, $\textcolor{red}{☐}$ must be $5$ to balance the equation

Let's simplify the equation, dealing with the fractions in both sides separately:

A. We'll start with the fraction on the left side:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}$Let's simplify this fraction while remembering the order of operations, meaning that exponents come before division and multiplication, which come before addition and subtraction, and parentheses come before everything,

Additionally, we'll remember that an exponent is multiplying a number by itself and therefore raising any number (even a negative number) to an even power will give a positive result, and this is because negative one multiplied by negative one gives the result of one,

In particular, in this problem:

$(-1)^8=1\\ (-3)^2=9$Let's return to the fraction and apply this, first we'll deal with the numerator of the fraction where we'll calculate the numerical value of the square root in the second term from the left and the value of the term with the exponent:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2}$Now we'll remember that according to the multiplication rules, multiplying a negative number by a negative number will give a positive result, we'll apply this to the numerator of the fraction in question, then we'll perform the multiplication in the second term from the left and simplify the expression in the numerator of the fraction:

$\frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{1+3\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{1+12+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{16}{6^2:(-3)^2-2^2\cdot2}$We'll continue and simplify the denominator of the fraction, we'll start by calculating the values of the terms with exponents, this we'll do, again, using what was said before (regarding the even exponent), then we'll perform the division operation in the first term from the left and the multiplication operation in the second term from the left, and finally we'll perform the subtraction operation in the denominator of the fraction:

$\frac{16}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{16}{36:9-4\cdot2}=\\ \frac{16}{4-8}=\\ \frac{16}{-4}=\\ -4$In the last stage we performed the division operation of the fraction, this while we remember that according to the division rules (which are identical to the multiplication rules in this context) dividing a positive number by a negative number will give a negative result,

We have finished simplifying the fraction on the left side, let's summarize this simplification:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\\ \frac{1-(-3)\cdot4+3}{6^2:(-3)^2-2^2\cdot2} =\\\ \frac{16}{6^2:(-3)^2-2^2\cdot2} =\\ \frac{16}{36:9-4\cdot2}=\\ \frac{16}{-4}=\\ -4$

B. Let's simplify the fraction on the right side:

$\frac{(-2)^3+\textcolor{red}{☐}^2-1}{-(-2)^2}$For convenience, let's name the number we're looking for and define it as x:

$\textcolor{red}{☐}=x\\ \downarrow\\ \textcolor{red}{☐}^2=x^2$Let's substitute this in the fraction in question:

$\frac{(-2)^3+x^2-1}{-(-2)^2}$So let's start simplifying the fraction, again we'll use the fact that raising any number to an even power will give a positive result, but we'll also remember the fact that raising a negative number to an odd power will give a negative result, in particular in the problem:

$(-2)^2=4\\ (-2)^3=-8$Let's apply this to the fraction in question:

$\frac{(-2)^3+x^2-1}{-(-2)^2} =\\ \frac{-8+x^2-1}{-4}$Now let's combine like terms in the numerator of the fraction:

$\frac{-8+x^2-1}{-4}=\\ \frac{-9+x^2}{-4}$We have finished simplifying the fraction on the right side of the given equation, let's summarize this simplification:

$\frac{(-2)^3+x^2-1}{-(-2)^2} =\\ \frac{-8+x^2-1}{-4} =\\ \frac{-9+x^2}{-4}$

Let's now return to the original equation and substitute the results of simplifying the fractions on the left and right sides detailed in A and B respectively, we won't forget the definition of the unknown x mentioned earlier:

$\frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\frac{(-2)^3+\textcolor{red}{☐}^2-1}{-(-2)^2} \\ \downarrow\\ \frac{(-1)^8-(-3)\cdot\sqrt{16}+3}{6^2:(-3)^2-2^2\cdot2}=\frac{(-2)^3+x^2-1}{-(-2)^2} \\ \downarrow\\ -4= \frac{-9+x^2}{-4}$ Let's continue and solve the resulting equation, first we'll remember that any number can be represented as itself divided by the number 1:

$-4= \frac{-9+x^2}{-4}\\ \frac{-4}{1}= \frac{-9+x^2}{-4}$Then we'll multiply both sides of the equation by the common denominator, which is the number $-4$ and then we'll simplify the equation and isolate the unknown by moving terms and combining like terms:

$\frac{-4}{1}= \frac{-9+x^2}{-4} \hspace{8pt}\text{/}\cdot(-4) \\ (-4)\cdot(-4)=-9+x^2\\ 16=-9+x^2\\ 16+9=x^2\\ 25=x^2$In the first stage we multiplied by the common denominator in order to cancel the fraction line while we multiply each numerator by the number that is the answer to the question "By how much did we multiply the current denominator to get the common denominator?" ,

Let's return to what was asked, we are looking for the number x (in the original problem it is a red square) for which we get a true statement from the last equation, we can take the square root of both sides of the equation and get the two possible solutions (we'll remember that taking a square root in the context of solving an equation always involves taking into account two possibilities, positive and negative) but we can also use logic and remember that raising any number to an even power will give a positive result, meaning the above equation has two possible answers, a positive answer and a negative answer,

We'll add and ask: "What number did we raise to the second power to get the number 25?" a question to which the answer is of course the number 5 (or minus 5), and therefore the full answer is:

$x=5,\hspace{4pt}-5$ Each of these options is correct,

And therefore the correct answer is answer C.

Let's simplify the expressions on both sides of the equation separately:

A. Let's start with the expression on the left side:

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)$Let's recall the order of operations in which powers take precedence over multiplication and division, and multiplication and division take precedence over addition and subtraction,

so let's start by simplifying the expressions inside the parentheses, where we will first calculate the values of the terms in the power and in the roots (which are powers of everything), then we will calculate the values of the products and results of the division operations, and finally we will perform the addition operations in the parentheses:

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2) =\\ (25+9):(4\cdot3+1+8:2) =\\ 34:(12+1+4)=\\ 34:17=\\ 2$In the last step we calculated the result of the division operation that remained (which was originally between the two main parentheses),

We have thus completed the simplification of the expression on the left side.

B. Let's continue simplifying the expression on the right side:

$\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}}$For convenience of operations, let's call this unknown number we are looking for, let's define it as- x:

$\textcolor{red}{☐}=x\\ \downarrow\\ \textcolor{red}{☐}^3=x^3\\$and we will place it in the mentioned expression:

$\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}} \\ \downarrow\\ \frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}}$Let's continue simplifying the expression, let's start by calculating their numerical value of the terms in the power and in the root in the numerator, in parallel we will calculate the numerical value of the term in the root in the denominator, then we will calculate the result of the subtraction operation in the numerator:

$\frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}}= \\ \frac{(36-4):x^3}{2} =\\ \frac{32:x^3}{2}$For convenience of solution for now, we will leave this expression in its current form and emphasize that we have completed the treatment of the expression on the right side.

Let's go back to the original equation and place in it the two simplification results we got for the terms on the left and right sides that were detailed in A and B, let's not forget that we also defined the number we are looking for as x:

$(5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)=\frac{(6^2-\sqrt{16}):\textcolor{red}{☐}^3}{\sqrt{4}} \\ \downarrow\\ (5^2+3^2):(\sqrt{16}\cdot\sqrt{9}+1+2^3:2)=\frac{(6^2-\sqrt{16}):x^3}{\sqrt{4}} \\ \downarrow\\ 2= \frac{32:x^3}{2} \\ \frac{2}{1}= \frac{32:x^3}{2}$$$In the last step we used the fact that any number can be written as a number divided by 1, we did this as a preparation for the next step where we will solve the equation that was obtained by multiplying both sides by the common denominator,

Now let's simplify the equation we got, again we will multiply both sides by the common denominator, but this time the common denominator is the algebraic expression: $x^3$, this expression depends on the unknown we are looking for and therefore we must define a permitted range of definition, since multiplying the equation by 0 is forbidden (and in general division by 0 is also a forbidden operation, so from the beginning we can define this range of definition):

$\frac{4}{1}=\frac{32}{x^3} \hspace{8pt}\text{/}\cdot x^3\hspace{4pt};x\neq0\\ 4 x^3=32$Again- we will know how much to multiply each term by using the answer to the question:"By how much did we multiply the current denominator in order to get the common denominator?", we also did not forget to define the permitted range of definition (we got it by solving the inequality:

$x^3\neq0$)

Let's continue and isolate the unknown by dividing both sides of the equation by its coefficient:

$4 x^3=32 \hspace{8pt}\text{/}:4\\ x^3=\frac{\not{32}}{\not{4}}\\ x^3=8$When in the last step we reduced the fraction that was obtained on the right side as a result of the division operation,

Let's finish solving the equation by taking the cube root of both sides of the equation, the cube root which is an operation inverse to the cube power will cancel the cube power on the unknown on the left side of the equation:

$x^3=8 \hspace{8pt}\text{/}\sqrt[3]{\hspace{4pt}}\\ \sqrt[3]{x^3}=\sqrt[3]{8}\\ x=2$Note that since we took an odd-order root from both sides of the equation we only need to consider one solution (it is the solution we will get in the calculator when connecting the cube root of the number 8) as opposed to taking an even-order root, in which case we need to consider two possible solutions - a positive solution and a negative solution,

We have thus solved the equation, and we got that the number we are looking for, which we defined to be the unknown x (marked at the beginning of the problem in a red square) is the number 2,

So the correct answer is answer D.

To solve the equation $\frac{(6^2-4^2):\sqrt{25}}{\sqrt{4}}=\frac{\sqrt{36}-\sqrt{49}:7}{5}+\textcolor{red}{☐}^{100}$, we need to simplify both sides step by step.

Let's start with the left-hand side (LHS):

• Calculate the powers: $6^2 = 36$ and $4^2 = 16$.
• Subtract the results: $36 - 16 = 20$.
• Calculate the square root: $\sqrt{25} = 5$.
• Perform the division: $20 : 5 = 4$.
• Calculate the square root in the denominator: $\sqrt{4} = 2$.
• Complete the division to simplify: $\frac{4}{2} = 2$.

So, the LHS simplifies to $2$.

Now, let's simplify the right-hand side (RHS):

• Calculate the square roots: $\sqrt{36} = 6$ and $\sqrt{49} = 7$.
• Perform the division inside the expression: $7:7 = 1$.
• Subtract the results: $6 - 1 = 5$.
• Divide by 5: $\frac{5}{5} = 1$.

So, the RHS simplifies to $1 + \textcolor{red}{☐}^{100}$.

This gives us the equation:

• $2 = 1 + \textcolor{red}{☐}^{100}$
• Solve for the missing number: $\textcolor{red}{☐}^{100} = 1$.

We know that $x^{100} = 1$ has two solutions for any real numbers: $x = 1$ and $x = -1$ because both $1^{100}$ and $(-1)^{100} = 1$ hold true.

Thus, the missing number is $1,\hspace{4pt}-1$.

Let's recall two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for an exponent applied to terms in parentheses:

$\big(\frac{a}{c}\big)^n=\frac{a^n}{c^n}$Unlike previous questions where we converted the square root to its corresponding half power according to the definition of root as an exponent, in this problem we will not make this conversion but rather understand and internalize that according to the definition of root as an exponent, the root is an exponent in every way and therefore all laws of exponents apply to it, particularly the law of exponents mentioned in b,

We will therefore use this understanding and simplify the expression in the left side, by applying the law of exponents mentioned in b to the first term:

$\sqrt{\frac{64}{10000}}+\frac{92}{10^2}=\textcolor{red}{☐}^{450} \\ \frac{\sqrt{64}}{\sqrt{10000}}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\ \frac{8}{100}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\$In the first stage, we applied the square root (which is actually a half power) to both numerator and denominator of the fraction in the first term on the left, this is according to the law of exponents mentioned in b and in the next stage we calculated the numerical value of the roots in the numerator and denominator of the fraction,

We'll continue and simplify the second term from the left on the left side, meaning we'll calculate the numerical value of the expression in the fraction's denominator, then we'll perform the addition operation between the two resulting fractions:

$\frac{8}{100}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\ \frac{8}{100}+\frac{92}{100} =\textcolor{red}{☐}^{450} \\ \frac{8+92}{100} =\textcolor{red}{☐}^{450} \\ \frac{100}{100} =\textcolor{red}{☐}^{450} \\ 1 =\textcolor{red}{☐}^{450}$In the second stage we performed the addition operation between the two fractions, using the fact that both denominators are identical (and therefore there was no need to expand them, but it was sufficient to combine them into one fraction using the identical denominator, which is actually the common denominator) then we simplified the expression in the numerator of the resulting fraction on the left side and finally remembered that dividing any number by itself will always give the result 1,

Now let's examine the equation we got in the last stage and answer the question that was asked,

On the left side we got the number 1, and on the right side there is a number (unknown) raised to the power of 450,

Remember that raising the number 1 to any power will always give the number 1, therefore the answer is the number 1,

Meaning:

$1 =\textcolor{red}{1}^{450}$However, since this is an even power (power of 450), we must also consider the negative possibility,

Meaning it also holds that:

$1=\textcolor{red}{(-1)}^{450}$

Therefore the correct answer is answer a.