To solve exponent problems, we need to apply the laws of exponents to simplify. Exponents allow us to write repeated multiplication compactly. By using the rules of exponents, we can facilitate the resolution of operations that involve several expressions with exponents.
Here, we will look at a summary of exponents along with the laws for exponents. We will also look at some examples with answers to facilitate the understanding of these types of exercises.
Summary of laws of exponents
A power is an expression that has the following form:
$latex {{b}^n}=b \times b … b \times b$
This represents the result of multiplying the base, b, by itself as many times as the exponent, n, indicates.
For example, $latex {{2}^4}=2 \times 2 \times 2 \times 2 = 16$ (the base is 2 and the exponent is 4).
We can have more complex expressions that combine different operations with exponents. The following are the laws of exponents, which tell us how to solve operations with powers. The letters a and b represent nonzero real numbers and the letters m and n represent whole numbers:
1) Law of zero exponents:
2) Law of negative exponents
3) Law of the product of exponents
4) Law of the quotient of exponents
5) Law of power of a power
6) Law of power of a product
7) Law of the quotient of a product
Examples with answers of exponent problems
The following exponent problems have their respective solution. The solution is detailed and indicates the process and reasoning used to obtain the answer. It is recommended that you try to solve the exercises yourself before looking at the answer.
EXAMPLE 1
Simplify the expression $latex {{5}^{-2}}$.
Solution
Here we have a negative exponent, so we have to start by converting that exponent to positive. Thus, applying the negative exponents’ rule, we have to take the reciprocal of the base and change the exponent from negative to positive:
$latex 5^{-2}=\frac{1}{5^2}$
Now, we apply the exponent to 5 to simplify:
$latex \frac{1}{{{5}^2}}=\frac{1}{25}$
EXAMPLE 2
Simplify the expression $latex {{({{3}^2})}^{-2}}$.
Solution
In this case, we have to apply the power of a power rule, where we multiply the exponents:
$latex {{({{3}^2})}^{-2}}={{3}^{-4}}$
Similar to the previous exercise, we apply the negative exponents’ rule to change the exponent from negative 4 to positive 4:
$latex {{3}^{-4}}=\frac{1}{{{3}^4}}$
Applying the exponent, we have:
$latex \frac{1}{{{3}^4}}=\frac{1}{81}$
EXAMPLE 3
Simplify the expression $latex \frac{{{4}^5}}{{{4}^3}}$.
Solution
We have a division of powers with the same base. Therefore, we use the quotient rule. This rule tells us that we subtract the exponent of the denominator from the exponent of the numerator:
$latex \frac{{{4}^5}}{{{4}^3}}={{4}^{5-3}}$
$latex ={{4}^2}$
Applying the exponent we have:
$latex {{4}^2}=16$
EXAMPLE 4
Simplify the expression $latex \frac{{{4}^5}\times {{3}^5}}{{{12}^3}}$.
Solution
In the numerator we have a multiplication of powers, but we cannot simplify because the base is not the same. However, we can rewrite the denominator by recognizing that 12 equals 4 × 3:
$latex \frac{{{4}^5}\times {{3}^5}}{{{12}^3}}=\frac{{{4}^5}\times {{3}^5}}{{{(4\times3)}^3}}$
$latex =\frac{{{4}^5}\times {{3}^5}}{{{4}^3}\times {{3}^3}}$
Now, we can apply the quotient rule to both base 3 powers and base 4 powers:
$latex \frac{{{4}^5}\times {{3}^5}}{{{4}^3}\times {{3}^3}}={{4}^{5-3}}\times {{3}^{5-3}}$
$latex ={{4}^2}\times {{3}^2}$
$latex =16\times 9$
$latex =144$
EXAMPLE 5
Simplify the expression $latex {{\left(\frac{4}{{{5}^2}} \right)}^{-2}} \times \left(\frac{{{4}^2}}{{{5}^3}} \right)$.
Solution
First, we can start by changing the negative exponent to positive by flipping the fraction over:
$latex {{\left(\frac{4}{{{5}^2}} \right)}^{-2}} \times \left(\frac{{{4}^2}}{{{5}^3}} \right)={{\left(\frac{{{5}^2}}{4} \right)}^2} \times \left(\frac{{{4}^2}}{{{5}^3}} \right)$
Now, we can apply the power of a power rule:
$latex {{\left(\frac{{{5}^2}}{4} \right)}^2} \times \left(\frac{{{4}^2}}{{{5}^3}} \right)=\frac{{{5}^4}}{{{4}^2}} \times \frac{{{4}^2}}{{{5}^3}} $
We can rewrite as follows and apply the quotient rule:
$latex \frac{{{5}^4}}{{{4}^2}} \times \frac{{{4}^2}}{{{5}^3}}=\frac{{{5}^4}}{{{5}^3}} \times \frac{{{4}^2}}{{{4}^2}}$
$latex ={{5}^{4-3}}\times{{4}^{2-2}}$
Simplifying and applying the zero rule of exponents, we have:
$latex {{5}^{4-3}}\times{{4}^{2-2}}={{5}^1}\times {{4}^0}$
$latex =5\times 1$
$latex =5$
EXAMPLE 6
Simplify the expression $latex \frac{{{x}^{-3}}{{y}^{2}}}{{{y}^2}{{x}^2}}$.
Solution
In this case, we have the variables x and y, but the rules of exponents are applied in the same way. Therefore, we start with the negative exponents’ rule:
$latex \frac{{{x}^{-3}}{{y}^2}}{{{y}^2}{{x}^2}}=\frac{{{y}^2}}{{{y}^2}{{x}^2}{{x}^3}}$
Now, we apply the quotient rule to the variable y and the product rule to the variable x:
$latex \frac{{{y}^2}}{{{y}^2}{{x}^2}{{x}^3}}=\frac{{{y}^{2-2}}}{{{x}^{2+3}}}$
$latex =\frac{1}{{{x}^5}}$
EXAMPLE 7
Simplify the expression $latex {{(7{{a}^3})}^{-2}}{{b}^{-1}}$.
Solution
Here, we can start with the negative exponents. Therefore, we take the reciprocal of the bases and change the exponents to positive:
$latex {{(7{{a}^3})}^{-2}}{{b}^{-1}}=\frac{1}{{{(7{{a}^3})}^2}{{b}^1}}$
Now, we apply the power of a power:
$latex \frac{1}{{{(7{{a}^3})}^2}{{b}^1}}=\frac{1}{{{7}^2}{{a}^6}{{b}^1}}$
$latex =\frac{1}{49{{a}^6}b}$
EXAMPLE 8
Simplify the expression $latex {{({{b}^{-3}}c)}^2}\times {{({{b}^{2}}{{c}^3})}^{-3}}$.
Solution
We start with the negative exponent on the right:
$latex {{({{b}^{-3}}c)}^2}\times {{({{b}^{2}}{{c}^3})}^{-3}}=\frac{{{({{b}^{-3}}c)}^2}}{{{({{b}^{2}}{{c}^3})}^3}}$
Now, we apply the power of a power rule to remove the parentheses:
$latex \frac{{{({{b}^{-3}}c)}^2}}{{{({{b}^{2}}{{c}^3})}^3}}=\frac{{{b}^{-6}}{{c}^2}}{{{b}^6}{{c}^9}}$
Again, we apply the negative exponents rule:
$latex \frac{{{b}^{-6}}{{c}^2}}{{{b}^6}{{c}^9}}=\frac{{{c}^2}}{{{b}^6}{{b}^6}{{c}^9}}$
Now, we apply the quotient rule to c and the product rule to b:
$latex \frac{{{c}^2}}{{{b}^6}{{b}^6}{{c}^9}}=\frac{1}{{{b}^{6+6}}{{c}^{9-2}}}$
$latex =\frac{1}{{{b}^{12}}{{c}^7}}$
Exponent problems – Practice
Apply what you have learned about powers and the rules of exponents with the following problems. Solve the problems and choose an answer. Check your answer to verify that you chose the correct answer.
See also
Interested in learning more about exponents? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
