Taking advantage of all the properties of powers or laws of exponents

From time to time, we will come across exercises in which we must use all the properties of powers together.
As soon as you have the exercise, try to first get rid of the parentheses according to the properties of powers and then, apply these properties to the corresponding terms, one after the other.

All the properties of powers or laws of exponents are:
am×an=a(m+n)a^m\times a^n=a^{(m+n)}
aman=a(mn)\frac {a^m}{a^n} =a^{(m-n)}
(a×b)n=an×bn(a\times b)^n=a^n\times b^n
(ab)n=anbn(\frac {a}{b})^n=\frac {a^n}{b^n}
(an)m=a(nm)(a^n )^m=a^{(n*m)}
a0=1a^0=1
When a0a≠0
an=1ana^{-n}=\frac {1}{a^n}

Practice Exponents Rules

Exercise #1

79×7= 7^9\times7=

Video Solution

Step-by-Step Solution

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:an×am=an+m a^n\times a^m=a^{n+m}

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

79+1=710 7^{9+1}=7^{10}

Answer

710 7^{10}

Exercise #2

828385= 8^2\cdot8^3\cdot8^5=

Video Solution

Step-by-Step Solution

All bases are equal and therefore the exponents can be added together.

828385=810 8^2\cdot8^3\cdot8^5=8^{10}

Answer

810 8^{10}

Exercise #3

1123=? \frac{1}{12^3}=\text{?}

Video Solution

Step-by-Step Solution

First, we recall the power property for a negative exponent:

an=1an a^{-n}=\frac{1}{a^n} We apply it to the expression we obtained:

1123=123 \frac{1}{12^3}=12^{-3} Therefore, the correct answer is option A.

Answer

123 12^{-3}

Exercise #4

8132= \frac{81}{3^2}=

Video Solution

Step-by-Step Solution

First, we recognize that 81 is a power of the number 3, which means that:

34=81 3^4=81 We replace in the problem:

8132=3432 \frac{81}{3^2}=\frac{3^4}{3^2} 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:

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

3432=342=32 \frac{3^4}{3^2}=3^{4-2}=3^2 Therefore, the correct answer is option b.

Answer

32 3^2

Exercise #5

2423= \frac{2^4}{2^3}=

Video Solution

Step-by-Step Solution

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:

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

2423=243=21 \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:

b1=b b^1=b Therefore, in the problem we obtain:

21=2 2^1=2 Therefore, the correct answer is option a.

Answer

2 2

Exercise #1

(4274)2= (\frac{4^2}{7^4})^2=

Video Solution

Step-by-Step Solution

(4274)2=42×274×2=4478 (\frac{4^2}{7^4})^2=\frac{4^{2\times2}}{7^{4\times2}}=\frac{4^4}{7^8}

Answer

4478 \frac{4^4}{7^8}

Exercise #2

(35)4= (3^5)^4=

Video Solution

Step-by-Step Solution

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

We use the property with our exercise and solve:

(35)4=35×4=320 (3^5)^4=3^{5\times4}=3^{20}

Answer

320 3^{20}

Exercise #3

42×44= 4^2\times4^4=

Video Solution

Step-by-Step Solution

To solve the exercise we use the property of multiplication of powers with the same bases:

anam=an+m a^n * a^m = a^{n+m}

With the help of this property, we can add the exponents.

42×44=44+2=46 4^2\times4^4=4^{4+2}=4^6

Answer

46 4^6

Exercise #4

192=? 19^{-2}=\text{?}

Video Solution

Step-by-Step Solution

To solve the exercise, we use the property of raising to a negative exponent

an=1an a^{-n}=\frac{1}{a^n}

We use the property to solve the exercise:

192=1192 19^{-2}=\frac{1}{19^2}

We can continue and solve the power

1192=1361 \frac{1}{19^2}=\frac{1}{361}

Answer

1361 \frac{1}{361}

Exercise #5

54×25= 5^4\times25=

Video Solution

Step-by-Step Solution

To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.

25=5 \sqrt{25}=5 25=52 25=5^2 Now, we go back to the initial exercise and solve by adding the powers according to the formula:

an×am=an+m a^n\times a^m=a^{n+m}

54×25=54×52=54+2=56 5^4\times25=5^4\times5^2=5^{4+2}=5^6

Answer

56 5^6

Exercise #1

(22)3+(33)4+(92)6= (2^2)^3+(3^3)^4+(9^2)^6=

Video Solution

Step-by-Step Solution

We use the formula:

(am)n=am×n (a^m)^n=a^{m\times n}

(22)3+(33)4+(92)6=22×3+33×4+92×6=26+312+912 (2^2)^3+(3^3)^4+(9^2)^6=2^{2\times3}+3^{3\times4}+9^{2\times6}=2^6+3^{12}+9^{12}

Answer

26+312+912 2^6+3^{12}+9^{12}

Exercise #2

(42)3+(g3)4= (4^2)^3+(g^3)^4=

Video Solution

Step-by-Step Solution

We use the formula:

(am)n=am×n (a^m)^n=a^{m\times n}

(42)3+(g3)4=42×3+g3×4=46+g12 (4^2)^3+(g^3)^4=4^{2\times3}+g^{3\times4}=4^6+g^{12}

Answer

46+g12 4^6+g^{12}

Exercise #3

(62)13= (6^2)^{13}=

Video Solution

Step-by-Step Solution

We use the formula:

(an)m=an×m (a^n)^m=a^{n\times m}

Therefore, we obtain:

62×13=626 6^{2\times13}=6^{26}

Answer

626 6^{26}

Exercise #4

(5x3)3= (5\cdot x\cdot3)^3=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)n=anbn (a\times b)^n=a^nb^n

(5×x×3)3=(15x)3 (5\times x\times3)^3=(15x)^3

(15x)3=(15×x)3 (15x)^3=(15\times x)^3

153x3 15^3x^3

Answer

153x3 15^3\cdot x^3

Exercise #5

(y×x×3)5= (y\times x\times3)^5=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)n=anbn (a\times b)^n=a^nb^n

(y×x×3)5=y5x535 (y\times x\times3)^5=y^5x^53^5

Answer

y5×x5×35 y^5\times x^5\times3^5