Problem 86 - Elementary Algebra

Question

Simplify \(\left(x^{4} y^{3}\right)^{2}\left(x y^{2}\right)^{4}\).

Step-by-Step Solution

Answer

Question: Simplify the given expression. \((\left(x^{4} y^{3}\right)^{2}\left(x y^{2}\right)^{4}\) Answer: \(x^{12} y^{14}\)

Step 1: Apply the power of a product rule

To apply the power of a product rule on both factors, remember that \((ab)^n = a^n b^n\). So, we have: \((\left(x^{4} y^{3}\right)^{2}\left(x y^{2}\right)^{4} = (x^4)^2 (y^3)^2(x)^4(y^2)^4\)

Step 2: Apply the power of a power rule

To apply the power of a power rule, remember that \((a^n)^m = a^{nm}\). So, for each variable, we have: \((x^4)^2 (y^3)^2(x)^4(y^2)^4 = x^{4 \cdot 2} y^{3 \cdot 2} x^{1 \cdot 4} y^{2 \cdot 4}\)

Step 3: Simplify exponent values

Next, we simplify the exponents values as follows: \(x^{4 \cdot 2} y^{3 \cdot 2} x^{1 \cdot 4} y^{2 \cdot 4} = x^8 y^6 x^4 y^8\)

Step 4: Apply product of powers rule

To apply the product of powers rule, remember that \(a^n a^m = a^{n+m}\). So, for variable \(x\) and variable \(y\), we have: \(x^8 y^6 x^4 y^8 = x^{8 + 4} y^{6 + 8}\)

Step 5: Simplify the expression and write the final answer

Finally, we simplify the expression and write the final result: \(x^{8 + 4} y^{6 + 8} = \boxed{x^{12} y^{14}}\)

Key Concepts

Power of a Product Rule

The power of a product rule is an algebraic principle that helps you manage expressions involving exponents. It states that when you raise a product to a power, you can distribute the exponent to each factor in the product. This means if you have

• ext{If }(ab)^n \text{ then it is equal to } a^nb^n

So in the given expression \((x^4y^3)^2(xy^2)^4\),you can individually raise each factor within the parentheses to the respective powers. Thus, this rule simplifies \((x^4y^3)^2\) to \((x^4)^2(y^3)^2\)and \((xy^2)^4\) to \((x)^4(y^2)^4\).This step by step distribution of exponents helps in breaking down complex expressions into manageable parts, making the simplification process straightforward and less error-prone.

Power of a Power Rule

The power of a power rule is another essential concept in algebra simplification involving exponents. It is used when you have an exponent raised to another exponent. The rule states:

• ext{If }(a^n)^m \text{ then it simplifies to } a^{nm}.

In our exercise, each power is simplified as follows:

• \((x^4)^2\) becomes \(x^{4 \times 2}\), which simplifies to \(x^8\)
• \((y^3)^2\) becomes \(y^{3 \times 2}\), simplifying to \(y^6\)
• \((x)^4\) is simply \(x^4\)
• \((y^2)^4\) simplifies to \(y^{2 \times 4}\), which is \(y^8\)

This helps simplify each component individually, avoiding any potential mix-up of variables and exponents.

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. They are an efficient way to express and simplify repeated multiplication. In algebra, understanding how to manipulate exponents is crucial.For example here, the expression \(\left(x^4 y^3\right)^2\left(x y^2\right)^4\) implies that

• x is multiplied by itself 4+4+2 times
• y is multiplied by itself 3+3+4 times.

When you're comfortable with the rules governing exponents, such as those involving products and powers, you can handle more complex expressions with ease. Exponents turn potentially difficult calculations involving large numbers into simpler arithmetic, focusing on their power properties.

Product of Powers Rule

The product of powers rule is a simple yet powerful rule used to simplify expressions with the same base and different exponents. It states:

• \text{If }a^n \times a^m, \text{ then it is equal to }a^{n+m}.

This rule applied to our simplified terms \(x^8, x^4\) and \(y^6, y^8\) works as follows:Firstly, you combine the powers of x:

• x^8 \times x^4 = x^{8+4} = x^{12}

Secondly, you do the same with the powers of y:

• y^6 \times y^8 = y^{6+8} = y^{14}

This rule is essential for neatly simplifying large expressions into a single term with combined powers, as shown in our final answer \(x^{12}y^{14}\). It greatly reduces complexity in polynomial expansion scenarios.