Problem 56 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \left(4 a^{6} b^{8}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(8a^9b^{12}\).

Step 1: Simplify the Exponent

The expression given is \((4a^6b^8)^{3/2}\). First, distribute the exponent \(\frac{3}{2}\) to each term within the parentheses. This gives us \(4^{\frac{3}{2}}a^{6 \times \frac{3}{2}}b^{8 \times \frac{3}{2}}\).

Step 2: Calculate the Power of 4

Compute \(4^{\frac{3}{2}}\). The expression \(4^{\frac{3}{2}}\) can be rewritten as \((4^1)^{3/2} = (2^2)^{3/2}\). By using the property \((x^m)^n = x^{mn}\), \((2^2)^{3/2} = 2^{3} = 8\).

Step 3: Simplify the Exponents for a and b

For \(a\), calculate \(a^{6\times\frac{3}{2}} = a^{9}\). For \(b\), compute \(b^{8\times\frac{3}{2}} = b^{12}\).

Step 4: Combine Results

Combine all parts to get the simplified expression. Thus, the final simplified expression is \(8a^9b^{12}\).

Key Concepts

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. For example, if you have the expression \( a^6 \), it means that \( a \) is multiplied by itself 6 times: \( a \times a \times a \times a \times a \times a \). Exponents can be positive or negative, and they can also be fractions. Fractional exponents represent roots of numbers. For example, an exponent of \( \frac{1}{2} \) means a square root, and an exponent of \( \frac{1}{3} \) means a cube root.
In the exercise, we see each part of the expression \((4a^6b^8)^{\frac{3}{2}}\) uses exponents. The exponent \( \frac{3}{2} \) is distributed across the numbers \(4\), \(a^6\), and \(b^8\), indicating a combination of squaring and cubing the results, which can be challenging but follows logically from understanding how exponents behave.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions into their simplest form. This often includes combining like terms or using mathematical operations to reduce the expression’s complexity.
In the example \((4a^6b^8)^{\frac{3}{2}}\), simplifying begins by distributing the exponents. Each component within the parentheses is raised to the power of \(\frac{3}{2}\). This requires us to rewrite \(4\) as \((2^2)\) and apply fractional exponents to \(a^6\) and \(b^8\).

• Calculate \( 4^{\frac{3}{2}} \) by rewriting it as \((2^2)^{\frac{3}{2}}\), which further simplifies to \(2^3\), giving \(8\).
• Convert \(a^6\) to \(a^{6 \times \frac{3}{2}} = a^9\).
• Change \(b^8\) to \(b^{8 \times \frac{3}{2}} = b^{12}\).

Once each part is simplified individually, they are combined in the end to form a single expression, \(8a^9b^{12}\).
The goal in simplifying expressions is to make them as straightforward and easy to use as possible without eliminating any critical components.

Properties of Exponents

Understanding the properties of exponents can greatly help in simplifying algebraic expressions. These properties offer rules that can save time and help avoid mistakes.
Some key properties include:

• Product of Powers Property: \(x^m \times x^n = x^{m+n}\).
• Power of a Power Property: \((x^m)^n = x^{mn}\).
• Power of a Product Property: \((xy)^n = x^n \times y^n\).
• Negative Exponent Property: \(x^{-n} = \frac{1}{x^n}\).
• Zero Exponent Property: \(x^0 = 1\) for any \(x eq 0\).

In our given problem, we apply the Power of a Product and Power of a Power Properties extensively. The expression \((4a^6b^8)^{\frac{3}{2}}\) utilizes these rules by distributing the fractional exponent across each term inside the parentheses. Applying these properties step by step leads us to the simplified expression \(8a^9b^{12}\).
Being familiar with these rules allows for a more intuitive approach when dealing with exponents, making algebraic manipulation easier and more efficient.