Question 222·Medium·Math

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The function $g$ is defined by $g(z)=6z^{-\tfrac{3}{2}}$ . What is the value of $g(9)$ ?

For function-evaluation questions with negative and fractional exponents, first substitute the given input into the function, then handle the exponent carefully: rewrite negative exponents using reciprocals ( $a^{-n}=1/a^n$ ) and fractional exponents using roots and powers (for example, $9^{3/2}=(9^{1/2})^3$ ). Simplify in small, clear steps and only at the end multiply by any coefficients, watching for common factors when reducing fractions.

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Start by substituting

Replace $z$ with $9$ in the expression $g(z)=6z^{-3/2}$ . What expression do you get for $g(9)$ ?

Understand the negative exponent

How can you rewrite $9^{-3/2}$ using a positive exponent? Recall that $a^{-n}=\dfrac{1}{a^n}$ .

Handle the fractional exponent

To evaluate $9^{3/2}$ , think of it as $(9^{1/2})^3$ . What is $9^{1/2}$ , and what do you get when you cube that result?

Finish the arithmetic

Once you find $9^{-3/2}$ , multiply that value by $6$ and simplify the resulting fraction.

Desmos Guide

Use Desmos to evaluate the expression

In a Desmos expression line, type exactly: 6*9^(-3/2) and look at the numeric output. Then match that decimal value to the equivalent fraction among the answer choices.

Step-by-step Explanation

Substitute the given value into the function

You are given the function $g(z)=6z^{-3/2}$ and asked for $g(9)$ .

Substitute $z=9$ into the formula:

g(9)=6\cdot 9^{-3/2}

Interpret the negative exponent

A negative exponent means "take the reciprocal." In general, $a^{-n}=\dfrac{1}{a^n}$ for nonzero $a$ .

Apply this idea to $9^{-3/2}$ :

9^{-3/2}=\frac{1}{9^{3/2}}

So now you have

g(9)=6\cdot \frac{1}{9^{3/2}}

Interpret the fractional exponent $3/2$

A fractional exponent like $9^{3/2}$ can be understood using roots and powers. One way is

9^{3/2} = \left(9^{1/2}\right)^3

Here, $9^{1/2}$ means the square root of $9$ , which is $3$ , so

9^{3/2}=3^3=27

Combine everything and simplify

From the previous step, $9^{3/2}=27$ , so

9^{-3/2}=\frac{1}{9^{3/2}}=\frac{1}{27}

Now plug this back into $g(9)$ :

g(9)=6\cdot \frac{1}{27}=\frac{6}{27}

Simplify $\frac{6}{27}$ by dividing numerator and denominator by their greatest common factor, $3$ :

\frac{6}{27}=\frac{2}{9}

So the value of $g(9)$ is $\boxed{\frac{2}{9}}$ .

Question 222·Medium·Math

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Step-by-step Explanation

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Step-by-step Explanation