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Chapter 5: Problem 138

Simplify each expression. $$ \left(b^{7}\right)^{5} \cdot\left(b^{2}\right)^{6} $$

Short Answer

Expert verified
The simplified expression is \(b^{47}\)

Step by step solution

01

- Apply the Power Rule

For the expression \(\big(b^{7}\big)^{5}\), apply the power rule \((a^m)^n = a^{m \times n}\) to simplify it. This gives \((b^{7})^5 = b^{35}\).
02

- Apply the Power Rule to the Second Term

Now, apply the power rule to the expression \(\big(b^{2}\big)^{6}\). Simplify it as follows: \((b^{2})^6 = b^{12}\).
03

- Multiply the Simplified Expressions

Combine the results from steps 1 and 2. According to the multiplication rule for exponents \(a^m \times a^n = a^{m+n}\), you get \((b^{35} \times b^{12}) = b^{35+12} = b^{47}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
To simplify expressions involving exponents, the power rule is amazingly helpful. The power rule states that when you raise a power to another power, you simply multiply the exponents.
For instance, consider the expression \(\big(b^{7}\big)^{5}\). According to the power rule, you multiply the exponents: \((b^7)^5 = b^{7 \times 5} = b^{35}\).
This technique makes it much easier to handle complex exponent operations. Similarly, applying the power rule to \(\big(b^{2}\big)^{6}\) gives \((b^2)^6 = b^{2 \times 6} = b^{12}\).
Following the power rule is important whenever you see an exponent raised to another exponent.
Multiplication Rule
After simplifying each part using the power rule, the multiplication rule for exponents is your next tool. The multiplication rule states that when you multiply two expressions with the same base, you add their exponents.
Let's use our previous results: we have \(b^{35} \times b^{12}\). By the multiplication rule, you add the exponents: \((b^{35} \times b^{12}) = b^{35+12} = b^{47}\).
This rule is powerful because it further simplifies complex expressions quickly and efficiently. Remember always to use this rule when you're dealing with multiplication of the same base.
Simplification
Simplification in mathematics helps to reduce expressions to their simplest form, making them easier to understand and work with.
In our example, we started with \(\big(b^{7}\big)^{5} \times \big(b^{2}\big)^{6}\) and simplified it step by step. First, we used the power rule to get \(b^{35}\) and \(b^{12}\). Next, we applied the multiplication rule to combine these results into \(b^{47}\).
Always follow a structured method: Identify which rules to apply first, simplify each part, and then combine the results. This approach makes even the most complicated expressions more manageable.

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Most popular questions from this chapter

Square each binomial using the Binomial Squares Pattern. $$ \left(8 p^{3}-3\right)^{2} $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (8 j+4)(8 j-4) $$

Multiply the binomials. Use any method. $$ (6 p+5)(p+1) $$

Square each binomial using the Binomial Squares Pattern. $$ \left(x+\frac{2}{3}\right)^{2} $$

Multiply the binomials using (a) the Distributive Property; (b) the FOIL method; (c) the Vertical Method. $$ (y+9)(y+3) $$
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