Problem 554 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(16^{\frac{3}{2}}\) (b) \(8^{\frac{2}{3}}\) (c) \(10,000^{\frac{3}{4}}\)

Step-by-Step Solution

Answer

a) 64, b) 4, c) 1,000.

Step 1: Simplify Part (a) – Identify the Expression

The given expression is \(16^{\frac{3}{2}}\).

Step 2: Simplify Part (a) – Apply Power Rule

Rewrite the expression using the power rule: \(a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}\) or \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Here, \(16 = 2^4\) so we have \[(16^{\frac{3}{2}} = (2^4)^{\frac{3}{2}} = 2^{4 \cdot \frac{3}{2}} = 2^6)\].

Step 3: Simplify Part (a) – Final Calculation

Calculate the power of 2: \(2^6 = 64\). Therefore, \(16^{\frac{3}{2}} = 64\).

Step 4: Simplify Part (b) – Identify the Expression

The given expression is \(8^{\frac{2}{3}}\).

Step 5: Simplify Part (b) – Apply Power Rule

Rewrite the expression using the power rule. Here, \(8 = 2^3\) so we have \[(8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2)\].

Step 6: Simplify Part (b) – Final Calculation

Calculate the power of 2: \(2^2 = 4\). Therefore, \(8^{\frac{2}{3}} = 4\).

Step 7: Simplify Part (c) – Identify the Expression

The given expression is \(10,000^{\frac{3}{4}}\).

Step 8: Simplify Part (c) – Apply Power Rule

Rewrite the expression using the power rule. Here, \(10,000 = 10^4\) so we have \[(10,000^{\frac{3}{4}} = (10^4)^{\frac{3}{4}} = 10^{4 \cdot \frac{3}{4}} = 10^3)\].

Step 9: Simplify Part (c) – Final Calculation

Calculate the power of 10: \(10^3 = 1,000\). Therefore, \(10,000^{\frac{3}{4}} = 1,000\).

Key Concepts

Power Rule

The power rule in exponents allows us to simplify expressions where a number is raised to a fractional power. The rule states that for any base \(a\), and integers \(m\) and \(n\), the expression \(a^{\frac{m}{n}}\) can be rewritten as either \(\sqrt[n]{a^m}\) or \((a^m)^{\frac{1}{n}}\). This property is extremely useful when working with fractional exponents, as it allows breaking down more complex expressions into more manageable parts.
For example, in the problem \(16^{\frac{3}{2}}\):

• First, rewrite 16 as a power of 2: \(16 = 2^4\).
• Next, apply the power rule: \((2^4)^{\frac{3}{2}}\).
• Simplify the exponents by multiplication: \(2^{4 \cdot \frac{3}{2}} = 2^6\).
• Calculate the final value: \(2^6 = 64\).

This step-by-step approach can be applied to any base and fractional exponent.

Fractional Exponents

Fractional exponents can sometimes seem intimidating, but understanding their meaning simplifies many tasks in algebra. A fractional exponent like \( \frac{m}{n} \) indicates both a power and a root. Specifically, \( a^{\frac{m}{n}} \) means we first raise \(a\) to the power of \(m\), and then take the \(n\)-th root of the result.
Consider the expression \(8^{\frac{2}{3}}\) from the exercise:

• Write 8 as a power of 2: \(8 = 2^3\).
• Apply the power rule: \( (2^3)^{\frac{2}{3}} \).
• Simplify by multiplying the exponents: \(2^{3 \cdot \frac{2}{3}} = 2^2\).
• Calculate the result: \(2^2 = 4\).

Understanding fractional exponents helps in simplifying expressions and solving equations involving roots and powers more easily.

Exponentiation

Exponentiation involves raising a base number to a given power. This operation is foundational in algebra and appears frequently in various mathematical contexts. When dealing with high powers and fractional exponents, knowing exponent rules is crucial.
For the expression \(10,000^{\frac{3}{4}}\):

• Rewrite 10,000 as a power of 10: \(10,000 = 10^4\).
• Apply the power rule: \((10^4)^{\frac{3}{4}}\).
• Multiply the exponents: \(10^{4 \cdot \frac{3}{4}} = 10^3\).
• Calculate the final value: \(10^3 = 1,000\).

Mastering exponentiation helps in understanding the behavior of functions, solving exponential equations, and simplifying large expressions, providing a solid foundation for more advanced math topics.