Problem 34 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(8 x^{2} y^{3}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression (8x^2y^3)^2. Answer: 64x^4y^6

Step 1: Identify the bases and exponent

We are given an expression (8x^2y^3)^2. Here, the bases are 8, x^2, and y^3, and the entire expression to the power of 2.

Step 2: Apply the power rule

Using the power rule `(a*b^c)^n = a^n * b^cn`: $$ \begin{aligned} \left(8x^2y^3\right)^2 &= 8^2 * (x^2)^2 * (y^3)^2 \end{aligned} $$

Step 3: Calculate powers

Now, we will compute the powers for each base: $$ \begin{aligned} 8^2 * (x^2)^2 * (y^3)^2 &= 64 * x^4 * y^6 \end{aligned} $$

Step 4: Write the simplified expression

Finally, we have simplified the given expression to: $$ 64x^4y^6 $$

Key Concepts

Power Rules for Exponents

Exponents are essential to simplify expressions, especially when handling variables. One core concept is the power rule which states that when you raise a power to another power, you multiply the exponents. For example,

• \((x^m)^n = x^{m \cdot n}\)
• \((a^m b^n)^p = a^{m \cdot p} b^{n \cdot p}\)

This is helpful when breaking down expressions involving powers.
When applied to a problem like \(\left(8x^2y^3\right)^2\), it becomes \(8^2 \cdot (x^2)^2 \cdot (y^3)^2\). Here, each part is separately raised to the power, making it simpler to handle each base individually.

Simplification of Expressions with Exponents

Simplification is the process of converting complex expressions into more manageable ones. When simplifying expressions with exponents, using the right rules helps minimize errors.

• Always apply the power rule to distribute the power to every term.
• Then, compute the powers of the constants separately from the variable expressions.

For instance, in \(\left(8x^2y^3\right)^2\), first calculate the power of 8: \(8^2 = 64\). Next, simplify \((x^2)^2\) and \((y^3)^2\):

• \((x^2)^2 = x^{2 \cdot 2} = x^4\)
• \((y^3)^2 = y^{3 \cdot 2} = y^6\)

This step-by-step approach ensures that the expression \(64x^4y^6\) is fully simplified.

Natural Numbers in Exponents

Natural numbers, which are positive integers starting from 1, play a crucial role in exponents. These numbers ensure that the powers are clearly defined and non-fractional, which keeps calculations straightforward. When dealing with natural number exponents:

• The operations are consistent and predictable.
• It prevents negative results or undefined expressions due to zero or fraction exponents.

In the expression \(\left(8x^2y^3\right)^2\), each power, such as 2 or 3, is a natural number, ensuring that every computation is valid and results in positive exponents, leading to an accurate simplification process. Understanding that natural numbers guide the exponent's behavior helps in anticipating the outcome of exponentiation.