Problem 108 - Elementary and Intermediate Algebra

Question

Use a calculator to evaluate each expression. Round to the nearest hundredth. See Using Your Calculator: Rational Exponents. $$ (-1,000)^{3 / 5} $$

Step-by-Step Solution

Answer

The value of \((-1,000)^{3/5}\) is approximately \(-63.10\).

Step 1: Understand the Expression

The expression to evaluate is \((-1,000)^{3/5}\). This involves a negative base \(-1,000\) and a rational exponent \(\frac{3}{5}\).

Step 2: Rewrite the Expression for Calculation

The expression \((-1,000)^{3/5}\) can be broken down as \([(-1,000)^{1/5}]^3\). The rational exponent \(\frac{3}{5}\) indicates taking the fifth root first, then raising the result to the third power.

Step 3: Calculate the Fifth Root

Find the fifth root of \(-1,000\). Use your calculator to find the fifth root (equivalently, the power of \(0.2\)) of \(-1,000\). This will result in approximately \(-3.98\).

Step 4: Raise to the Third Power

Now raise the result of the fifth root, \(-3.98\), to the third power using your calculator: \((-3.98)^3 = -63.1\) after rounding to the nearest hundredth.

Step 5: Verify and Round the Answer

Ensure your calculations are correct by re-evaluating the calculation steps. Confirm the final answer is rounded to the nearest hundredth as \(-63.10\).

Key Concepts

Fifth Root Calculation

Calculating fifth roots might seem tricky at first, but with a calculator, it becomes much easier. The fifth root of a number is the value that, when multiplied by itself five times, will give the original number. In mathematical terms, for a number \(x\), its fifth root is \(x^{1/5}\). When dealing with negative numbers, like \(-1,000\), the process is the same, but one must ensure the calculator can handle complex roots. For instance, inputting \(-1,000^{1/5}\) in a calculator will often require a special function to assure correct handling of the negative base, due to the intricacies of complex numbers. It's equivalent to finding the power of \(0.2\). The result in this specific case is approximately \(-3.98\).

Negative Bases in Exponents

Working with negative bases and exponents requires understanding how the signs affect the outcome. With a negative base like \(-1,000\), raising it to a rational exponent affects the sign of the result. Even though negative exponents can make calculations a bit harder, especially with fractions, the principles remain the same. It's critical to realize how the power and root alter the sign; an odd root, like the fifth root, will keep the negative sign, such as turning \(-1,000\) into \(-3.98\). If you raise a negative base to an integer exponent and it's odd, the result is negative. Therefore, when raising \(-3.98\) to the third power, you still end up with a negative result, resulting in approximately \(-63.1\).

Rounding Decimals

Rounding decimals is a key skill when presenting calculations, especially when the result is lengthy. Rounding involves looking at the digit beyond where you want to round off. Here, rounding to the nearest hundredth means observing the third decimal place.

• If it's 5 or more, round the preceding hundredth place up by one.
• If it's less than 5, the hundredth place remains the same.

This rule ensures precision and ease of reading without excess digits. For example, the result of \(-63.104\) gets rounded to \(-63.10\), because the thousandth place is less than 5. Rounding provides a neat, readable format of your solutions, making it crucial for accurately presenting homework assignments or quick estimates.