Problem 536 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(81^{\frac{1}{2}}\) (b) \(125^{\frac{1}{3}}\) (c) \(64^{\frac{1}{2}}\)

Step-by-Step Solution

Answer

a) 9, b) 5, c) 8

Step 1 - Understanding the Problem

The problem involves simplifying expressions with fractional exponents. Recall that a fractional exponent such as \(\frac{1}{2}\) or \(\frac{1}{3}\) represents a root. For example, \(a^{\frac{1}{2}}\) represents the square root of \(a\), and \(a^{\frac{1}{3}}\) represents the cube root of \(a\).

Step 2 - Simplify Part (a)

Simplify \(81^{\frac{1}{2}}\). The exponent \(^{\frac{1}{2}}\) indicates the square root. Therefore, \(\begin{aligned}81^{\frac{1}{2}} = \sqrt{81} = 9 \end{aligned}\).

Step 3 - Simplify Part (b)

Simplify \(125^{\frac{1}{3}}\). The exponent \(^{\frac{1}{3}}\) indicates the cube root. Therefore, \(\begin{aligned}125^{\frac{1}{3}} = \sqrt[3]{125} = 5 \end{aligned}\).

Step 4 - Simplify Part (c)

Simplify \(64^{\frac{1}{2}}\). The exponent \(^{\frac{1}{2}}\) indicates the square root. Therefore, \(\begin{aligned}64^{\frac{1}{2}} = \sqrt{64} = 8 \end{aligned}\).

Key Concepts

simplifying expressions

Simplifying expressions is an essential skill in algebra and beyond. It makes complex-looking problems easy to understand and solve.

For example, consider expression simplifications involving fractional exponents. A fractional exponent like \(\frac{1}{2}\) or \(\frac{1}{3}\) can be simplified by understanding their root equivalents.

• \(\frac{1}{2}\) indicates the square root of a number.
• \(\frac{1}{3}\) indicates the cube root of a number.

To simplify \[81^{\frac{1}{2}}\], we know the \(\frac{1}{2}\) exponent corresponds to the square root. So, \[81^{\frac{1}{2}} = \sqrt{81} = 9.\] Similarly, for cube roots, \[125^{\frac{1}{3}} = \sqrt[3]{125} = 5.\] By breaking down these expressions using fractional exponents, simplifying becomes straightforward.

square roots

Square roots are a specific type of root that tells us what number multiplied by itself gives us the original number. For example, \(9 \times 9 = 81\), so the square root of 81 is 9.

When we see an expression like \[81^{\frac{1}{2}}\], it signifies the square root of 81. The notation \(\frac{1}{2}\) as an exponent tells us to find this root: \[81^{\frac{1}{2}} = \sqrt{81} = 9.\]

The square root operation can be summarized as looking for output that, when squared, gives you back the initial input. This principle is used widely in algebra, geometry, and real-world applications like calculating areas.

cube roots

Cube roots work similarly to square roots but involve finding a number that, when cubed (multiplied by itself twice more), results in the original number. For example, since \(5 \times 5 \times 5 = 125\), the cube root of 125 is 5.

In expressions, a cube root is represented by a \(\frac{1}{3}\) exponent. For example: \[125^{\frac{1}{3}} = \sqrt[3]{125} = 5. \] This tells us that 5, when multiplied by itself three times, equals 125.

Understanding cube roots is crucial, especially in tasks involving volumes and higher- degree polynomial equations. It is a fundamental concept that enables us to break down and simplify complex algebraic expressions efficiently.