Applying Combined Exponents Rules - Examples, Exercises and Solutions

We use the zero exponent rule.

$X^0=1$We obtain

$112^0=1$Therefore, the correct answer is option C.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information

• Step 2: Apply the appropriate exponent rule

• Step 3: Perform the calculation

Now, let's work through each step:
Step 1: The problem gives us the expression $1^3$. This means we have a base of 1 and an exponent of 3.
Step 2: We'll use the exponentiation rule, which states that $a^n = a \times a \times \ldots \times a$ (n times).
Step 3: Since our base is 1, raising 1 to any power will still result in 1. Therefore, we can express this as $1 \times 1 \times 1 = 1$.

Therefore, the solution to $1^3$ is $1$.

To solve the problem of finding $7^0$, we will follow these steps:

• Step 1: Identify the general rule for exponents with zero.

• Step 2: Apply the rule to the given problem.

• Step 3: Consider the provided answer choices and select the correct one.

Now, let's work through each step:

Step 1: A fundamental rule in exponents is that any non-zero number raised to the power of zero is equal to one. This can be expressed as: $a^0 = 1$ where $a$ is not zero.

Step 2: Apply this rule to the problem: Since we have $7^0$, and $7$ is certainly a non-zero number, the expression evaluates to 1. Therefore, $7^0 = 1$.

Therefore, the solution to the problem is $7^0 = 1$, which corresponds to choice 2.

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$Remember that any number raised to the 1st power is equal to the number itself, meaning that:

$b^1=b$Therefore, in the problem we obtain:

$2^1=2$Therefore, the correct answer is option a.

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{3^5}{3^2} = 3^{5-2}$

Simplifying, we get $3^3$

To solve this problem, we'll employ the power of a power rule in exponents, which states that $(a^m)^n = a^{m \times n}$.

Let's apply this rule to each part of the expression:

• Step 1: Simplify $(3^2)^4$
  According to the power of a power rule, this becomes $3^{2 \times 4} = 3^8$.

• Step 2: Simplify $(5^3)^5$
  Similarly, apply the rule here to get $5^{3 \times 5} = 5^{15}$.

After simplifying both parts, we multiply the results:

$3^8 \times 5^{15}$

Thus, the reduced expression is $\boxed{3^8 \times 5^{15}}$.

In order to solve this problem, we'll follow these steps:

• Step 1: Identify the base and exponents

• Step 2: Use the formula for multiplying powers with the same base

• Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is $(3^4) \times (3^2)$. Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
$a^m \times a^n = a^{m+n}$

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
$3^4 \times 3^2 = 3^{4+2} = 3^6$

Therefore, the simplified form of the expression is $3^6$.

To solve this problem, let's follow these steps:

• Understand the zero exponent rule.

• Apply this rule to the given expression.

• Identify the correct answer from the given options.

According to the rule of exponents, any non-zero number raised to the power of zero is equal to $1$. This is one of the fundamental properties of exponents.
Now, apply this rule:

Step 1: We are given the expression $(-3)^0$.
Step 2: Here, $-3$ is our base. We apply the zero exponent rule, which tells us that $(-3)^0 = 1$.

Therefore, the value of $(-3)^0$ is $1$.

To solve the exercise we use the power property:$(a^n)^m=a^{n\cdot m}$

We use the property with our exercise and solve:

$(3^5)^4=3^{5\times4}=3^{20}$

To simplify the given expression $4^2 \times 3^5 \times 4^3 \times 3^2$, we will follow these steps:

• Step 1: Identify and group similar bases.

• Step 2: Apply the rule for multiplying like bases.

• Step 3: Simplify the expression.

Now, let's go through each step thoroughly:

Step 1: Identify and group similar bases:
We see two distinct bases here: 4 and 3.

Step 2: Apply the rule for multiplying like bases:
For base 4: Combine $4^2$ and $4^3$, using the rule $a^m \times a^n = a^{m+n}$.

Add the exponents for base 4: $2 + 3 = 5$, thus, $4^2 \times 4^3 = 4^5$.

For base 3: Combine $3^5$ and $3^2$, still using the same exponent rule.

Add the exponents for base 3: $5 + 2 = 7$, resulting in $3^5 \times 3^2 = 3^7$.

Step 3: Simplify the expression:
The simplified expression is $4^5 \times 3^7$.

Therefore, the final simplified expression is $4^5 \times 3^7$.

To solve this problem, we'll follow these steps:

• Step 1: Identify and group the terms with the same base.

• Step 2: Apply the laws of exponents to simplify by adding the exponents of each base.

• Step 3: Write the simplified form.

Let's work through each step:

Step 1: We are given that $4^7 \times 5^3 \times 4^2 \times 5^4$.

Step 2: First, group the terms with the same base:

$4^7 \times 4^2$ and $5^3 \times 5^4$.

Step 3: Use the law of exponents, which states $a^m \times a^n = a^{m+n}$.

For the base 4: $4^7 \times 4^2 = 4^{7+2} = 4^9$.

For the base 5: $5^3 \times 5^4 = 5^{3+4} = 5^7$.

Therefore, the simplified form of the expression is $4^9 \times 5^7$.

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{5^6}{5^4} = 5^{6-4}$. Simplifying, we get $5^2$.

Let's simplify the expression $5^3 \times 2^4 \times 5^2 \times 2^3$ using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

• Step 1: Focus on terms with the same base.
  Combine $5^3$ and $5^2$. Since both terms have the base $5$, we apply the rule $a^m \times a^n = a^{m+n}$: $5^3 \times 5^2 = 5^{3+2} = 5^5$

• Step 2: Combine $2^4$ and $2^3$. Similarly, for the base $2$: $2^4 \times 2^3 = 2^{4+3} = 2^7$

After simplification, the expression becomes:
$5^5 \times 2^7$

To solve this problem, we'll follow these steps:

• Identify the given information and relevant exponent rules.

• Apply the quotient property of exponents.

• Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression $\frac{6^7}{6^4}$. The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ In this case, $a = 6$, $m = 7$, and $n = 4$.
Step 3: Applying this rule gives us: $\frac{6^7}{6^4} = 6^{7 - 4} = 6^3$

Therefore, the solution to the problem is $6^3$.

To simplify the equation $6^4 \times 2^3 \times 6^2 \times 2^5$, we will make use of the rules of exponents, specifically the product of powers rule, which states that when multiplying two powers that have the same base, you can add their exponents.

Step 1: Identify and group the terms with the same base.
In the expression $6^4 \times 2^3 \times 6^2 \times 2^5$, group the powers of 6 together and the powers of 2 together:

• Powers of 6: $6^4 \times 6^2$

• Powers of 2: $2^3 \times 2^5$

Step 2: Apply the product of powers rule.
According to the product of powers rule, for any real number $a$, and integers $m$ and $n$, the expression $a^m \times a^n = a^{m+n}$.

Apply this rule to the powers of 6:
$6^4 \times 6^2 = 6^{4+2} = 6^6$.

Apply this rule to the powers of 2:
$2^3 \times 2^5 = 2^{3+5} = 2^8$.

Step 3: Write down the final expression.
Combining our results gives the simplified expression: $6^6 \times 2^8$.

Therefore, the solution to the problem is $6^6 \times 2^8$.