Questions: Use the definition of rational exponents to write each of the following with the appropriate root. Then simplify: (144/25)^(1/2)=

Transcript text: Use the definition of rational exponents to write each of the following with the appropriate root. Then simplify: \[ \left(\frac{144}{25}\right)^{\frac{1}{2}}= \]

Solution

Solution Steps

To solve the given problem, we need to use the definition of rational exponents. A rational exponent of \(\frac{1}{2}\) indicates taking the square root of the base. Therefore, we need to find the square root of the fraction \(\frac{144}{25}\). This can be done by taking the square root of the numerator and the denominator separately.

Step 1: Apply the Definition of Rational Exponents

We start with the expression \(\left(\frac{144}{25}\right)^{\frac{1}{2}}\). According to the definition of rational exponents, this can be rewritten as the square root of the fraction: \[ \left(\frac{144}{25}\right)^{\frac{1}{2}} = \frac{\sqrt{144}}{\sqrt{25}} \]

Step 2: Simplify the Square Roots

Next, we simplify the square roots of the numerator and the denominator: \[ \sqrt{144} = 12 \quad \text{and} \quad \sqrt{25} = 5 \] Thus, we can substitute these values back into our expression: \[ \frac{\sqrt{144}}{\sqrt{25}} = \frac{12}{5} \]

Step 3: Convert to Decimal Form

To express \(\frac{12}{5}\) in decimal form, we perform the division: \[ \frac{12}{5} = 2.4 \]