Problem 52 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(2 a^{3} b^{2}\right)^{2}\left(5 a^{2} b^{5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( 500a^{12}b^{19} \).

Step 1: Expand the Powers

Firstly, apply the power of a power rule \( (x^m)^n = x^{m \cdot n} \) separately to each part of the expression:1. \( (2a^3b^2)^2 = 2^2 \cdot (a^3)^2 \cdot (b^2)^2 = 4a^6b^4 \).2. \( (5a^2b^5)^3 = 5^3 \cdot (a^2)^3 \cdot (b^5)^3 = 125a^6b^{15} \).Now the expression becomes \( 4a^6b^4 \cdot 125a^6b^{15} \).

Step 2: Combine Like Terms

Now, simplify by multiplying the coefficients and adding the exponents of the same base:* For the coefficients: \( 4 \times 125 = 500 \).* For \( a \): \( a^6 \times a^6 = a^{6+6} = a^{12} \).* For \( b \): \( b^4 \times b^{15} = b^{4+15} = b^{19} \).So the simplified expression is \( 500a^{12}b^{19} \).

Step 3: Final Simplized Expression

The simplified expression \( 500a^{12}b^{19} \) is simplified completely as all powers are positive. Thus, the elimination of negative exponents is already achieved in the simplification.

Key Concepts

Laws of Exponents

Exponents are a shorthand way to express repeated multiplication. The laws of exponents help us to simplify expressions involving powers more easily. Here are some key rules:

• Power of a Power Rule: This states that \( (x^m)^n = x^{m \cdot n} \). It means that when you raise a power to another power, you multiply the exponents.
• Product of Powers Rule: When multiplying two powers with the same base, you simply add their exponents: \( x^m \cdot x^n = x^{m+n} \).
• Power of a Product Rule: This tells us that when raising a product to a power, we can distribute the power to each factor in the product, like \( (xy)^n = x^n y^n \).

In the exercise, the laws of exponents are used to simplify complex expressions by applying the power of a power rule to each component and then using the product of powers rule to simplify the entire expression. Understanding these laws makes it easier to manipulate and simplify algebraic terms.

Simplifying Expressions

Simplifying an expression involves making it easier to work with by reducing it to its simplest form. The goal is to present the expression with the least number of terms and factors possible, without changing its value.

To simplify expressions:

• Apply the Laws of Exponents: Use exponent rules to condense multiplication and division of like bases.
• Combine Like Terms: Add or subtract terms that have identical variable parts. For example, \( a^6 \cdot a^6 = a^{12} \).
• Simplify Coefficients: Multiply or divide numerical coefficients separately from variables.

In the given problem, after expanding the powers using the power of a power rule, like terms were combined by adding exponents for similar bases, while coefficients were multiplied to present the simplest form.

Multiplying Polynomials

Multiplying polynomials involves distributing each term of one polynomial to every term of the other polynomial. In general, you need to apply the distributive property repeatedly.

Key steps include:

• Distribute Each Term: Multiply each term in the first polynomial by each term in the second polynomial.
• Combine Like Terms: After distribution, collect and simplify terms that have the same variable parts.
• Simplify: Apply the laws of exponents where possible to further simplify the expression.

In our exercise, polynomial multiplication happens in two stages: expanding each polynomial raised to a power and finally multiplying the resulting expressions. It demonstrates how polynomial multiplication can be enhanced by exponent laws to yield a clean and concise solution.