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89. If [tex]\left(x^2\right)^y = x^{16}[/tex], then which of the following is equivalent to [tex]y[/tex]?

A) 4
B) 6
C) 8
D) 14


Sagot :

To solve the equation [tex]\(\left(x^2\right)^y = x^{16}\)[/tex], we need to use properties of exponents. Let's go through the steps in detail:

1. Apply the power of a power property to the left side of the equation:

The property [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex] allows us to rewrite [tex]\(\left(x^2\right)^y\)[/tex]:
[tex]\[ \left(x^2\right)^y = x^{2 \cdot y} \][/tex]

2. Rewrite the equation using the exponent property:

Substituting the left side of the original equation, we get:
[tex]\[ x^{2y} = x^{16} \][/tex]

3. Since the bases are the same, set the exponents equal to each other:

For the equation [tex]\( x^{a} = x^{b} \)[/tex] to be true, [tex]\(a\)[/tex] must equal [tex]\(b\)[/tex]. Therefore, we have:
[tex]\[ 2y = 16 \][/tex]

4. Solve for [tex]\(y\)[/tex]:

Isolate [tex]\(y\)[/tex] by dividing both sides of the equation by 2:
[tex]\[ y = \frac{16}{2} \][/tex]
[tex]\[ y = 8 \][/tex]

Thus, the value of [tex]\(y\)[/tex] is [tex]\(8\)[/tex]. Therefore, the answer is:
[tex]\[ \boxed{8} \][/tex]