Problem 556 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(27^{\frac{5}{3}}\) (b) \(16^{\frac{5}{4}}\) (c) \(32^{\frac{2}{5}}\)

Step-by-Step Solution

Answer

(a) 243, (b) 32, (c) 4

Step 1: Simplify the exponent in part (a)

Rewrite the base 27 as a power of 3: \[ 27 = 3^3 \] Then use the property of exponents \((a^m)^n = a^{m \times n}\): \[ 27^{\frac{5}{3}} = (3^3)^{\frac{5}{3}} \] Now multiply the exponents: \[ (3^3)^{\frac{5}{3}} = 3^{3 \times \frac{5}{3}} = 3^5 \] Finally, calculate \(3^5\): \[ 3^5 = 243 \]

Step 2: Simplify the exponent in part (b)

Rewrite the base 16 as a power of 2: \[ 16 = 2^4 \] Then use the property of exponents \((a^m)^n = a^{m \times n}\): \[ 16^{\frac{5}{4}} = (2^4)^{\frac{5}{4}} \] Now multiply the exponents: \[ (2^4)^{\frac{5}{4}} = 2^{4 \times \frac{5}{4}} = 2^5 \] Finally, calculate \(2^5\): \[ 2^5 = 32 \]

Step 3: Simplify the exponent in part (c)

Rewrite the base 32 as a power of 2: \[ 32 = 2^5 \] Then use the property of exponents \((a^m)^n = a^{m \times n}\): \[ 32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} \] Now multiply the exponents: \[ (2^5)^{\frac{2}{5}} = 2^{5 \times \frac{2}{5}} = 2^2 \] Finally, calculate \(2^2\): \[ 2^2 = 4 \]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions efficiently. An exponent tells you how many times to multiply the base by itself. For instance, in the expression \(a^m\), 'a' is the base and 'm' is the exponent.

There are several important properties, such as:

• \((a^m)^n = a^{mn}\): When raising a power to another power, you multiply the exponents.
• \(a^m \times a^n = a^{m+n}\): When multiplying like bases, you add the exponents.
• \(a^{-n} = \frac{1}{a^n}\): A negative exponent indicates a reciprocal.

Using these properties simplifies the calculation and helps you handle even complex expressions with ease.

Simplifying Exponents

Simplifying exponents involves breaking down the base into simpler components and then applying the properties of exponents.

For example, to simplify \(27^{\frac{5}{3}}\), we first recognize that 27 can be written as \(3^3\). Then we apply the power-to-power property \((a^m)^n = a^{mn}\):

\[(3^3)^{\frac{5}{3}} = 3^{3 \times \frac{5}{3}} = 3^5\]

Finally, we calculate \(3^5 = 243\).

• Start by expressing the base as a power of a smaller base, if possible.
• Apply the properties of exponents methodically.
• Perform the arithmetic to obtain the simplified form.

This systematic approach ensures you don’t miss any steps and reach the correct answer efficiently.

Base and Exponent Relationship

The base and exponent relationship indicates how the base is repeatedly multiplied. The base is the number you are multiplying, while the exponent tells you how many times to multiply it by itself.

For instance, in the term \(2^5\), 2 is the base and 5 is the exponent, meaning we multiply 2 by itself 5 times:

\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\]

Understanding this relationship allows you to manipulate and simplify expressions more intuitively. Changing the expression 27 into \(3^3\) and then using the exponent \(\frac{5}{3}\) harnesses this relationship and makes the simplification straightforward. Recognizing how a number can be broken down into its prime factors and then raised to a given power is a powerful strategy in algebra and higher-level mathematics.