Problem 50 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 4. $$ \left(27 a^{3} b^{3}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( 9a^2b^2 \).

Step 1: Apply the Power Rule

The expression given is \( \left(27 a^{3} b^{3}\right)^{2/3} \). Using the power rule \( (x^m)^n = x^{m \cdot n} \), apply the exponent \( \frac{2}{3} \) to each factor inside the parentheses separately. Thus, it becomes \( 27^{2/3} (a^3)^{2/3} (b^3)^{2/3} \).

Step 2: Simplify the Numerical Component

Calculate \( 27^{2/3} \). The cube root of 27 is 3, because \( 3^3 = 27 \). Raise this result to the power of 2: \( 3^2 = 9 \). Thus, \( 27^{2/3} = 9 \).

Step 3: Simplify the Variable a Component

For the variable part \( (a^3)^{2/3} \), use the power rule to multiply the exponents: \( 3 \cdot \frac{2}{3} = 2 \). Therefore, \( (a^3)^{2/3} = a^2 \).

Step 4: Simplify the Variable b Component

Similarly, for \( (b^3)^{2/3} \), multiply the exponents: \( 3 \cdot \frac{2}{3} = 2 \). Therefore, \( (b^3)^{2/3} = b^2 \).

Step 5: Combine Simplified Components

Combine the simplified components from Steps 2, 3, and 4: \( 9 \times a^2 \times b^2 \). This gives the simplified expression: \( 9a^2b^2 \).

Key Concepts

Power Rules and Their Simplification

The Power Rule is a handy tool for working with exponents, often written as:

• \((x^m)^n = x^{m \cdot n}\).

This formula tells us that when raising a power to another power, you multiply the exponents. Let's break it down a bit more!
Imagine you have a number raised to an exponent, like

• \((a^3)^2\).

Instead of calculating

• \(a^3 \times a^3\),

we use the Power Rule to make it simpler:

• \(a^{3 \times 2}\) which equals \(a^6\).

By using the Power Rule, you can save time and effort, particularly when dealing with larger exponents. This principle can be applied to all similar expressions, making calculations much faster.

Understanding Cube Root

The cube root is the number that, when multiplied by itself three times, gives you the original number. If a number is written

• \(x^{1/3}\),

it means the cube root of \(x\).
A practical example: To find the cube root of 27, we're looking for a number that when multiplied by itself three times equals 27. That's 3, because:

• \(3 \times 3 \times 3 = 27\).

Thus, the cube root of 27 is 3. Notice how simple it can be! This principle helps towards simplifying entire expressions where numbers and variables are raised to fractional exponents.
Keep in mind that cube roots can also apply to integers raised to a power, helping simplify complex expressions previously hard to crack.

Explore the World of Exponents

Exponents serve as a shorthand way of expressing repeated multiplication.
Let's take

• \(a^3\),

which means \(a\) multiplied by itself twice,

• \(a \times a \times a\).

When exponents come into play, you can represent very big or very small numbers easily. In algebra, they are useful for showing powers of variables.
For example,

• \((b^3)^{2/3}\)

is a variable powered by another fraction.
How do we approach this? Multiply the inner exponent by the outer fraction:

• \(3 \times \frac{2}{3} = 2\).

So, it becomes

• \(b^2\).

When learning algebra, grasping the basics of exponents can open up new possibilities and help you solve complex problems more comfortably!