Applying Combined Exponents Rules - Examples, Exercises and Solutions

Understanding Applying Combined Exponents Rules

Complete explanation with examples

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}

Detailed explanation

Practice Applying Combined Exponents Rules

Test your knowledge with 44 quizzes

Simplify the following equation:

\( 7^5\times2^3\times7^2\times2^4= \)

Examples with solutions for Applying Combined Exponents Rules

Step-by-step solutions included
Exercise #1

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

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

Video Solution
Exercise #2

9993= \frac{9^9}{9^3}=

Step-by-Step Solution

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

bmbn=bmn \frac{b^m}{b^n}=b^{m-n}

Let's apply it to the problem:

9993=993=96 \frac{9^9}{9^3}=9^{9-3}=9^6

Therefore, the correct answer is b.

Answer:

96 9^6

Video Solution
Exercise #3

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

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}

Video Solution
Exercise #4

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

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}

Video Solution
Exercise #5

Solve the exercise:

(a5)7= (a^5)^7=

Step-by-Step Solution

We use the formula:

(am)n=am×n (a^m)^n=a^{m\times n}

and therefore we obtain:

(a5)7=a5×7=a35 (a^5)^7=a^{5\times7}=a^{35}

Answer:

a35 a^{35}

Video Solution

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