Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(32^{4 / 5}\) means the fourth _____ of the fifth _____ of 32.

Step-by-Step Solution

Answer

power, root

Step 1: Understand the Exponent

The expression \(32^{4/5}\) represents a fractional exponent. A fractional exponent like \(a^{m/n}\) means to take the nth root of \(a\) and then raise it to the mth power.

Step 2: Identify the Fifth Root

The denominator of the fractional exponent, which is 5 in this case, indicates the root we need to calculate. Therefore, the fifth root of 32 is calculated as 32 to the power of \(1/5\), or simply the root which results in 2, since \(2^5 = 32\).

Step 3: Calculate the Fourth Power

Once we have the fifth root of 32 as 2, we raise it to the power given by the numerator, which is 4 in this case. Thus, we calculate \(2^4 = 16\).

Step 4: Fill in the Blanks

Using the breakdown of the fractional exponent, the expression \(32^{4/5}\) is the fourth power of the fifth root of 32.

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication. When you see something like \(a^n\), it means you multiply \(a\) by itself \(n\) times. This is a shortcut to express such operations, which is extremely useful in mathematics. For example, \(3^2\) means \(3 \times 3 = 9\). Exponents help simplify mathematical expressions by reducing long multiplication into a concise form.

With exponential notation, you can express complex calculations succinctly, which is especially beneficial when dealing with large numbers or advanced calculations. They play a crucial role in many areas of mathematics and science.

The Concept of Roots

Roots are essentially the opposite of exponents. While exponents deal with repeated multiplication, roots address repeated division. The most common root is the square root, which can be represented as \(\sqrt{a}\) or \(a^{1/2}\). This means finding a number that, when multiplied by itself, gives you the original number \(a\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

Another example is the cube root, represented as \(a^{1/3}\), which means finding a number that multiplied by itself three times gives the original number. Understanding roots is crucial for working with fractional exponents, as seen in the expression \(32^{4/5}\), where the fifth root is determined first.

Interpreting Mathematical Expressions

Mathematical expressions use a combination of numbers, variables, and operators to represent a value or equation. In the case of fractional exponents, such as \(32^{4/5}\), these expressions combine both exponents and roots to deliver a precise mathematical instruction.

In this expression, the denominator of the fractional exponent tells you the root to take first. Here, 5 represents the fifth root. Then, the numerator, 4 in this case, indicates the power to which the root result should be raised. This two-step process allows the expression to be evaluated accurately, demonstrating the harmony between exponents and roots in mathematical problem-solving.

• First, find the root: interpret the denominator.
• Second, raise to the power: use the numerator.

Understanding how to translate these symbols into actionable steps is key when working with complex mathematical expressions.