Problem 133 - College Algebra

Question

In Exercises 132–135, determine whether each statement makes sense or does not make sense, and explain your reasoning. If \(5^{-2}\) is raised to the third power, the result is a number between 0 and 1

Step-by-Step Solution

Answer

Yes, the statement makes sense. If \(5^{-2}\) is raised to the third power, the result is indeed a number between 0 and 1.

Step 1: Understanding Negative Exponents

A negative exponent means the base, which is 5 in this case, is on the wrong side of the fraction line, so you can start by moving it to the other side. So, \(5^{-1} = 1/5\) and \(5^{-2} = 1/(5^2) = 1/25\).

Step 2: Calculating Powers of Powers

When you are raising a power to a power, you multiply the exponents. In this case, \( (5^{-2})^3 = 5^{-6}\). Now move the 5 to the other side of the fraction line to get 1/15625 because \(5^6=15625\).

Step 3: Evaluate the Result

The result, \( 1/15625 \), is a number between 0 and 1, as 0 < \(1/15625\) < 1 .

Key Concepts

Understanding Negative Exponents

Grasping the concept of negative exponents is crucial for mastering algebra. Picture a negative exponent as an instruction to 'flip' the base to the other side of a fraction. For instance, consider the expression with a base of 5 and a negative exponent: \(5^{-2}\). How do you interpret this? It's simpler than you might think: any number with a negative exponent means you take the reciprocal of the number, raised to the positive of that exponent. In this case, \(5^{-2}\) becomes \(\frac{1}{5^2}\) which simplifies to \(\frac{1}{25}\).

This 'reciprocal rule' is a fundamental principle in algebra that clears the confusion around negative exponents. It's like saying, 'Instead of multiplying by 5 twice, I'll divide by 5 twice.' And hence, \(5^{-2}\) is indeed a small number between 0 and 1 because it represents a fraction of the whole.

Multiplying Powers of Powers

When it comes to 'powers of powers,' we venture into multiplying exponents, which is another vital concept in algebra. But don't worry, there's a very straightforward rule for it. If you have a power raised to another power, like \((5^{-2})^3\), you multiply the exponents. This is where the mathematics of it gets fun—you're essentially taking the power operation to another level.

Why do you multiply? Because you are applying the repeated multiplication implied by the exponent multiple times. For example, \(5^{-2}\) means \(\frac{1}{5} \times \frac{1}{5}\), and when raised to the third power \((5^{-2})^3\), you're multiplying \(5^{-2}\) by itself three times. The 'power to a power' rule tells you to multiply -2 by 3, giving you an exponent of -6. Therefore, \((5^{-2})^3 = 5^{-6} = \frac{1}{5^6} = \frac{1}{15625}\), reinforcing that this expression's value is a positive number between 0 and 1.

Fractional Exponents and Their Interpretation

The world of exponents isn't limited to whole numbers. Fractional exponents bring in even more versatility. A fractional exponent, such as \(5^{\frac{1}{2}}\), encapsulates both exponentiation and rooting in one notation. It tells us that we need to take the square root of 5, since the denominator of the fraction is 2.

For a general case, the expression \(x^{\frac{m}{n}}\) means taking the nth root of x and then raising it to the mth power, or vice versa. It's interchangeable. You might be working through a problem and come across a pesky fraction in the exponent. Remember, don't panic! It's just another way of expressing a root or a radical. Calculations with these exponents obey the same laws as integral exponents. This means you can multiply and divide them—and even combine them with powers of powers—using the same rules you know and love from basic exponents.