Problem 43 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(-7 a^{2} b^{5} c\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(49 a^{4} b^{10} c^{2}\).

Step 1: Apply the power to each term

The expression we need to simplify is \((-7 a^{2} b^{5} c)^{2}\). According to the power of a product rule, we need to apply the power to each term inside the parenthesis: \((-7)^{2} (a^{2})^{2} (b^{5})^{2} (c)^{2}\).

Step 2: Simplify the coefficient

Compute \((-7)^{2}\). Since squaring a negative number results in a positive number, we get \((-7)^{2} = 49\).

Step 3: Simplify the power of each variable

Use the power rule for exponents to simplify the expression for each variable:- For \(a^{2}\), apply the power rule: \((a^{2})^{2} = a^{2 \times 2} = a^{4}\).- For \(b^{5}\), apply the power rule: \((b^{5})^{2} = b^{5 \times 2} = b^{10}\).- For \(c\), apply the power rule: \((c)^{2} = c^{2}\).

Step 4: Combine the results

Combine all the terms together to form the simplified expression: \(49 a^{4} b^{10} c^{2}\).

Key Concepts

Power of a Product

The power of a product refers to the rule in mathematics where you apply the exponent to each factor within the parentheses separately. In simple terms, when you have something like \((xy)^{n}\), you distribute the exponent to both \(x\) and \(y\), resulting in \(x^{n}y^{n}\). This rule is particularly helpful when dealing with complex algebraic expressions.
Let's relate this to our original exercise: \((-7 a^{2} b^{5} c)^{2}\). You need to apply the power of 2 to each term inside the parentheses step by step:

• Apply \(2\) to \(-7\), making it \((-7)^{2}\).
• Apply \(2\) to \(a^{2}\), resulting in \((a^{2})^{2}\).
• Apply \(2\) to \(b^{5}\), resulting in \((b^{5})^{2}\).
• Lastly, apply \(2\) to \(c\), resulting in \(c^{2}\).

Once applied, you simplify each part to transform the original expression into an easier version, which is what we ultimately achieved in the last step of the solution.

Exponentiation

Exponentiation is the process of raising a number or an expression to the power of another number. It is essentially repeated multiplication. For instance, raising \(a\) to the power of \(b\) denotes multiplying \(a\) by itself \(b\) times, written as \(a^{b}\).
In our example, we are working with \((-7a^{2}b^{5}c)^{2}\). Each variable or coefficient is raised to the power of 2 as part of exponentiation. For example:

• The coefficient \(-7\) is squared to get \(49\), since \((-7) \times (-7) = 49\).
• Next, \(a^{2}\) squared becomes \(a^{4}\) since \((a^{2})^{2} = a^{2 \times 2}\).
• Similarly, \(b^{5}\) raised to the power 2 becomes \(b^{10}\), as shown by \((b^{5})^{2} = b^{5 \times 2}\).
• Finally, \(c\) squared is simply \(c^{2}\).

Exponentiation helps in handling such algebraic expressions efficiently, simplifying complex calculations by transforming them into fewer steps.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and mathematical operations. These expressions are fundamental in algebra and can include terms like \(2x + 3\) or \(5a^{2} - 4a + 7\).
In our problem, the algebraic expression is \((-7 a^{2} b^{5} c)^{2}\). It consists of multiple terms and involves the application of exponents. Such expressions often require simplification, as was done with our exercise.

• Each variable term had to be handled using appropriate rules like the power of a product.
• The coefficients needed to be calculated straightforwardly using exponentiation.
• After applying these steps, we arrived at:\(49 a^{4} b^{10} c^{2}\) - a concise version of the original expression.

Understanding and working with algebraic expressions is crucial as it forms the basis for more advanced algebraic operations and mathematical problem-solving.