Select the correct answer.
Which expression is equivalent to the given expression?
$\frac{6 a b}{\left(a^0 b^2\right)^4}$
A. $\frac{6}{a^3 b^6}$
B. $\frac{6 a}{b^7}$
C. $\frac{6}{a^3 b^7}$
D. $\frac{6 a}{b^5}$
Answer (1)
Simplify the denominator using the rule a 0 = 1 and the power of a power rule: ( a 0 b 2 ) 4 = ( b 2 ) 4 = b 8 .
Rewrite the expression as a fraction: b 8 6 ab .
Simplify the fraction by dividing the numerator and denominator by b : b 8 6 ab = b 7 6 a .
The equivalent expression is b 7 6 a .
Explanation
Understanding the Problem We are given the expression ( a 0 b 2 ) 4 6 ab and asked to find an equivalent expression. Let's simplify the given expression step by step.
Simplifying the Denominator First, we simplify the denominator. Recall that any non-zero number raised to the power of 0 is 1, so a 0 = 1 . Thus, the denominator becomes ( 1 ⋅ b 2 ) 4 = ( b 2 ) 4 .
Applying the Power of a Power Rule Next, we use the power of a power rule, which states that ( x m ) n = x m ⋅ n . Applying this rule, we have ( b 2 ) 4 = b 2 ⋅ 4 = b 8 . So the expression becomes b 8 6 ab .
Simplifying the Fraction Now, we simplify the fraction by dividing b by b 8 . Recall that b n b m = b m − n . Thus, b 8 b = b 1 − 8 = b − 7 . So the expression becomes 6 a b − 7 .
Rewriting with Positive Exponent Finally, we rewrite b − 7 as b 7 1 . Therefore, the expression becomes b 7 6 a .
Selecting the Correct Option Comparing our simplified expression b 7 6 a with the given options, we see that it matches option B.
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating the area and volume of geometric shapes. For example, if you have a rectangular prism with dimensions involving exponents, simplifying the expressions allows you to easily find the volume. Suppose the length is 2 x , the width is 3 x 2 , and the height is x − 1 . The volume would be V = ( 2 x ) ( 3 x 2 ) ( x − 1 ) = 6 x 2 + 1 − 1 = 6 x 2 . This skill is also useful in physics when dealing with units and scientific notation.
