Problem 4 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ 25^{-3 / 2} $$

Step-by-Step Solution

Answer

The expression \(25^{-3/2}\) evaluates to \(\frac{1}{125}\).

Step 1: Understand the Expression

The expression given is \(25^{-3/2}\). This means we have a base of 25 raised to the power of \(-3/2\). This power is a combination of a negative exponent and a fractional exponent.

Step 2: Apply Rule of Negative Exponents

A negative exponent implies that we take the reciprocal of the base. Therefore, \(25^{-3/2} = \frac{1}{25^{3/2}}\).

Step 3: Understand Fractional Exponents

The fractional exponent \(\frac{3}{2}\) means we first take the square root and then raise the result to the third power, or vice versa. Essentially, \(b^{m/n} = (\sqrt[n]{b})^m\).

Step 4: Evaluate the Square Root

Calculate the square root of 25. Since \(\sqrt{25} = 5\), we have \((25^{1/2}) = 5\).

Step 5: Raise to the Third Power

Raise the result from Step 4 to the power of 3: \(5^3 = 125\).

Step 6: Complete the Reciprocal

Since \(\frac{1}{25^{3/2}} = \frac{1}{125}\), this is the value of our expression.

Key Concepts

Negative Exponents

When you encounter an expression with a negative exponent, it can be a bit confusing at first. But don't worry, it's actually quite simple. A negative exponent means you take the reciprocal of the base. In other words, you "flip" the base upside down. For example, if you have a base of 25 with an exponent of -1, it becomes \( \frac{1}{25} \).

• The negative sign doesn't affect the base directly but tells you to invert it.
• Keep the exponent's magnitude as is, while inverting the base.

This is crucial when dealing with fractional exponents, which we'll discuss next.

Fractional Exponents

Fractional exponents might seem tricky, but they are just another way to represent roots and powers. You might spot them in the form of \( b^{m/n} \). The denominator \( n \) of the fraction indicates the root, while the numerator \( m \) tells you the power.

• For \( \frac{3}{2} \), you first find the square root (because \( 2 \) is in the denominator).
• Then, you use the result to calculate the power (raise it to the "3" in the numerator).

This means \( 25^{3/2} \) can be seen as \((\sqrt{25})^3\) or equivalently \( \sqrt{25^3} \). Both paths will give the same result, ensuring you're comfortable with roots and powers.

Square Roots

Square roots are based on the idea of finding a number which when multiplied by itself gives the original number. It's commonly denoted by the symbol \( \sqrt{} \). For example, \( \sqrt{25} \) asks, what number times itself equals 25? The answer is 5.

• Square roots are part of the process in simplifying expressions with fractional exponents.
• Remember that \( \sqrt{25} \) equals 5 because \( 5 \times 5 = 25 \).

Square roots are fundamental in math and understanding them clears a path for breaking down complex exponent tasks. Think of them as essential stepping stones in solving problems like the one in our exercise.