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What is the value of [tex]\( b \)[/tex] in the equation below?

[tex]\[ \frac{5^6}{5^2} = a^b \][/tex]

A. 3
B. 4
C. 5
D. 8


Sagot :

To solve the equation [tex]\(\frac{5^6}{5^2} = a^b\)[/tex], let's follow these steps:

1. Simplify the left-hand side:

We start with the expression [tex]\(\frac{5^6}{5^2}\)[/tex]. Using the properties of exponents, specifically the quotient rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we can simplify:

[tex]\[ \frac{5^6}{5^2} = 5^{6-2} = 5^4 \][/tex]

2. Compare the simplified left-hand side to the right-hand side:

After simplifying, we have:

[tex]\[ 5^4 = a^b \][/tex]

In this equation, it's clear that [tex]\(a = 5\)[/tex] because the base on both sides must be the same.

3. Determine the value of [tex]\(b\)[/tex]:

Now we have:

[tex]\[ 5^4 = 5^b \][/tex]

Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore:

[tex]\[ b = 4 \][/tex]

Hence, the value of [tex]\(b\)[/tex] is [tex]\(4\)[/tex].