Problem 28 - Beginning and Intermediate Algebra

Question

Simplify. $$(4 y)^{2}(3 y)^{3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(432y^5\).

Step 1: Apply the power of a power rule

In the given expression, we have two terms raised to powers, and we need to apply the power of a power rule. The power of a power rule states that \((a^m)^n = a^{m\times n}\). Applying this rule to our expression, we get: \((4y)^2(3y)^3 = (4^2 \times y^2)(3^3 \times y^3)\)

Step 2: Combine like terms

Now, we combine the constants and the variables separately: \(4^2\times 3^3 \times y^2 \times y^3\)

Step 3: Multiply constants and apply the product of power rule

Multiply the constants and apply the product of power rule, which states that \(a^m\times a^n = a^{m+n}\): \(16 \times 27 \times y^{2+3} = 432y^5\) So, the simplified form of the given expression is \(432y^5\).

Key Concepts

Exponents

Exponents are mathematical notations that indicate the number of times a number, called the base, is multiplied by itself. In the expression \((4y)^2(3y)^3\), **exponents** are used to show power operations applied to the bases (in this case, \(4y\) and \(3y\)). An exponent in the form \(a^n\) means you multiply the base \(a\) by itself \(n\) times.\
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For example, in \(3^3\), the base is 3, and the exponent is 3. This translates to multiplying 3 by itself three times: \(3 \times 3 \times 3 = 27\). Likewise, in \(y^3\), \(y\) is multiplied by itself three times. Therefore, understanding **exponents** is crucial as they streamline expressions and play a key role in algebra.

Simplifying Expressions

Simplifying expressions means transforming a complex expression into its simplest form. This involves combining like terms, performing arithmetic operations, and employing mathematical properties and rules. In the expression \((4y)^2(3y)^3\), we simplified it to \(432y^5\). \
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Here’s how simplification works:\

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• **Identifying like terms:** This involves recognizing components that can be combined, such as powers of the same base.\
• **Perform arithmetic:** We calculate the values of constants and multiply, divide, add, or subtract them as necessary.\
• **Use algebraic rules:** Using algebraic rules like the power of a product or sum helps to simplify.\

In this problem, the constants (4 and 3) were first modified using the power rule, then multiplied together. Similarly, the powers of \(y\) were combined using exponent rules to achieve the simplified expression. The process involves turning a seemingly complex problem into a more straightforward solution.

Rules of Exponents

The rules of exponents are essential tools in algebra for simplifying expressions involving exponential numbers. These rules make operations with exponents more manageable and less tedious.\
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Key rules applied in the exercise include:\

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• **Power of a power rule:** This rule states that \((a^m)^n = a^{m \cdot n}\). It’s pivotal when raising an expression to another power, as we did from \((4y)^2\) to \(4^2y^2\).\
• **Product of powers rule:** This rule posits that \(a^m \times a^n = a^{m+n}\), used when combining like bases. In the exercise, we used it to simplify \(y^2 \times y^3\) to \(y^{2+3} = y^5\).\

These rules help break down and reduce expressions to a simpler form efficiently. Using these right allows us to handle complicated expressions swiftly and correctly without losing accuracy in calculations.