Problem 134 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 5^{-2}>2^{-5} $$

Step-by-Step Solution

Answer

False. The correct statement is \(8^{-1/3} = 1/2\).

Step 1: Interpret the negative fraction exponent

Begin by recognizing that the negative fraction exponent invokes two rules: the rule of negative exponents and the rule of fractional exponents. When an exponent is negative, it means you're dealing with the reciprocal of the base, and when it's a fraction, it's alluding to a root. So, \(8^{-1/3}\) is equivalent to \(1/(8^{1/3})\) or the reciprocal of the cube root of 8.

Step 2: Calculate the cube root of 8

Proceed to find the cube root of 8, which is \(8^{1/3}\). The cube root of 8 is drawn out by checking what number times itself three times will equal to 8. The number that complies with this is 2, so \(8^{1/3} = 2\). Thus, our expression becomes \(1/2\).

Step 3: Compare the result with the statement

The result derived is \(1/2\), but in the original statement the number on the right side is \(-2\). These are not the same, which translates to the initial statement being false.

Step 4: Correct the statement

You rectify the original statement by replacing the \(-2\) on the right side with the result from step 2, which is \(1/2\). Hence, the correct statement becomes \(8^{-1/3} = 1/2\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with power and root operations. An exponent indicates how many times a base number is multiplied by itself. For example, when we see a number like \(5^3\), it tells us that the number 5 is multiplied by itself 3 times: \(5 \times 5 \times 5\).

However, things become slightly more complicated with negative and fractional exponents. With negative exponents, such as \(x^{-n}\), the rule states that you take the reciprocal of the base raised to the positive exponent, which means \(x^{-n} = \frac{1}{x^n}\). When you encounter a fractional exponent, like \(x^{\frac{1}{n}}\), it indicates a root. Specifically, \(x^{\frac{1}{n}}\) is the n-th root of x. Combining these two rules for negative fraction exponents, we find that for an expression like \(8^{-\frac{1}{3}}\), we are looking at the reciprocal of the cube root of 8.

Cube Root

The cube root of a number provides an answer to the question: 'What number, when multiplied by itself three times, gives the original number?' It's symbolized as \(\sqrt[3]{x}\) or \(x^{\frac{1}{3}}\).

For instance, \(\sqrt[3]{8}\) is asking for a number which when cubed, or multiplied by itself two more times (three in total), equals 8. The cube root of 8 is indeed 2, because \(2 \times 2 \times 2 = 8\). This operation is essential when dealing with fractional exponents like \(8^{-\frac{1}{3}}\), where we first find the cube root of 8 and then apply other relevant exponent rules.

Reciprocal of a Number

The reciprocal of a number is essentially a 'flipped' version of the number, where the numerator and denominator are swapped. For any non-zero number 'a', its reciprocal is \(\frac{1}{a}\).

If you're looking at integers, the reciprocal of 2 is \(\frac{1}{2}\), or if you have a fraction like \(\frac{3}{4}\), the reciprocal is \(\frac{4}{3}\). In the context of negative fraction exponents, such as \(8^{-\frac{1}{3}}\), we seek the reciprocal of the cube root of 8. So, after finding that the cube root of 8 is 2, we determine its reciprocal to be \(\frac{1}{2}\).

Rational Exponents

Rational exponents represent an exponent that is a fraction, where the numerator is a power and the denominator is a root. They are another way to express roots and powers succinctly. For example, \(x^{\frac{m}{n}}\) means 'the n-th root of x raised to the m-th power'.

This notation is particularly useful because it adheres to the usual rules of exponents, such as \(x^{a}x^{b} = x^{a+b}\), which simplifies working with complex expressions involving roots. For the exercise at hand, \(8^{-\frac{1}{3}}\) is a rational exponent where the cube root (denominator of 3) is taken first, followed by the reciprocal due to the negative sign—a clear application of rational exponent principles.