Problem 18 - Precalculus

Question

For the functions \(f(x)=3^{x}, g(x)=\left(\frac{1}{16}\right)^{x},\) and \(h(x)=10^{x+1},\) find the function value at the indicated points. $$g\left(-\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

64

Step 1: Understand the function structure

The function provided is \( g(x) = \left(\frac{1}{16}\right)^{x} \). This is an exponential function where the base is \( \frac{1}{16} \).

Step 2: Substitute the value for x

We need to evaluate \( g \left(-\frac{3}{2}\right) \) by substituting \(-\frac{3}{2}\) for \(x\). So, the expression becomes \( g \left(-\frac{3}{2}\right) = \left(\frac{1}{16}\right)^{-\frac{3}{2}} \).

Step 3: Apply the negative exponent rule

The negative exponent \(-\frac{3}{2}\) means we take the reciprocal of the base and raise it to the positive equivalent of the exponent. Thus, \( \left(\frac{1}{16}\right)^{-\frac{3}{2}} = \left(16\right)^{\frac{3}{2}} \).

Step 4: Convert fractional exponent to radical form

The expression \(16^{\frac{3}{2}}\) can be rewritten using radicals: \( \sqrt{16^3} \). This represents the square root of \(16\) raised to the third power.

Step 5: Calculate the exponentiation and radical

Start by computing \( \sqrt{16} \), which equals 4. Then raise 4 to the power of 3: \( 4^3 = 64 \).

Step 6: Conclusion

Therefore, \( g\left(-\frac{3}{2}\right) \) evaluates to 64.

Key Concepts

Negative Exponents

Negative exponents tell us to take the reciprocal of the base number and then raise it to the absolute value of the exponent. Let's simplify this concept:

When given a negative exponent, say \( a^{-n} \), it translates to \( \frac{1}{a^n} \). This means you're flipping the base \( a \) to its reciprocal and changing the exponent to a positive number.

In the function \( g(x) = \left( \frac{1}{16} \right)^{x} \), substituting \(-\frac{3}{2}\) for \(x\) results in \( \left( \frac{1}{16} \right)^{-\frac{3}{2}} \). You first need to transform the base using the negative exponent rule:

• Flip the fraction to \( 16 \).
• Change the negative exponent \( -\frac{3}{2} \) to a positive exponent \( \frac{3}{2} \).

This conversion results in \( 16^{\frac{3}{2}} \). Understanding this basic rule helps in simplifying expressions and solving more complex problems.

Fractional Exponents

Fractional exponents can seem tricky at first, but they are quite logical once broken down. A fractional exponent represents both a power and a root. For example, an expression like \( a^{\frac{m}{n}} \) indicates that you need to perform two steps:

• Take the \( n \)-root (radical) of \( a \).
• Raise the result to the \( m \)-th power.

In our problem, after converting \( \left( \frac{1}{16} \right)^{-\frac{3}{2}} \) to \( 16^{\frac{3}{2}} \):

• The exponent \( \frac{3}{2} \) means to first find the square root of \( 16 \) (the denominator indicates the root).
• Then, you raise the result to the third power (the numerator indicates the power).

This approach gives you more ways to visualize and compute expressions involving fractional exponents effectively.

Radicals

Radicals are another way to express the roots of numbers, and they work hand in hand with fractional exponents. When dealing with expressions like \( a^{\frac{1}{n}} \), this becomes the \( n \)-th root of \( a \), also written as \( \sqrt[n]{a} \).

In our solution, converting \( 16^{\frac{3}{2}} \) into a radical:

• Recognize that \( \frac{1}{2} \) in the fractional exponent denotes a square root.
• Begin by calculating the square root of 16, so it becomes \( \sqrt{16} \), which equals 4.

After that, proceed with the power aspect of the fractional exponent:

• Raise the result 4 to the power of 3: \( 4^3 \).
• This multiplication yields 64, providing the final solution.

Working through radicals and recognizing their transformation from fractional exponents can make solving these problems much more intuitive.