Problem 299 $$\text { (a) }q^{5} \cdot q^{3}... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 299

$$\text { (a) }q^{5} \cdot q^{3} \quad \text { (b) }\left(q^{5}\right)^{3}$$

Short Answer

Expert verified

(a) $q^{8}$, (b) $q^{15}$

Step by step solution

Simplify Part (a)

The first part requires us to simplify the expression $q^{5} \times q^{3}$. When multiplying expressions with the same base, we add the exponents. Thus,\[q^{5} \times q^{3} = q^{5+3} = q^{8}\]

Simplify Part (b)

The second part requires us to simplify the expression $\big(q^{5}\big)^{3}$. When raising a power to another power, we multiply the exponents. Thus,\[\big(q^{5}\big)^{3} = q^{5 \times 3} = q^{15}\]

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Key Concepts

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

Multiplying Exponents

Multiplying exponents might seem tricky at first, but it becomes simple once we understand the basic rule. When multiplying two expressions that have the same base, we simply add their exponents.
For instance, in the exercise, we have the expression $q^{5} \times q^{3}$. Here, both terms share the same base, which is $q$.
According to the rule, we add the exponents together:
$q^{5} \times q^{3} = q^{5+3} = q^{8}$
This rule holds true no matter how large the exponents are. As long as the bases are the same, just add the exponents.

Power of a Power Rule

The power of a power rule is another key concept for working with exponents. This rule applies when you have an exponent raised to another exponent.
For example, consider the expression from the exercise $\big(q^{5}\big)^{3}$.
This rule tells us to multiply the exponents together:
$\big(q^{5}\big)^{3} = q^{5 \times 3} = q^{15}$
By multiplying the exponents, we simplify the expression into a single power of $q$. This reduces complexity and makes calculations easier.
Remember, the base remains unchanged while we use multiplication to combine the exponents.

Simplifying Expressions with Exponents

Simplifying expressions with exponents is all about applying the rules accurately and reducing the expression to its simplest form.
First, identify if you are dealing with multiplication of exponents or a power raised to another power.
For multiplication, add the exponents:

Example: $a^{2} \times a^{3} = a^{2+3} = a^{5}$

For a power of a power, multiply the exponents:

Example: $\big(b^{4}\big)^{2} = b^{4 \times 2} = b^{8}$

Combining these rules lets you handle more complex expressions efficiently.
Practice these steps regularly to become proficient in simplifying expressions with exponents.

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91影视

$$\text { (a) }q^{5} \cdot q^{3} \quad \text { (b) }\left(q^{5}\right)^{3}$$

Short Answer

Step by step solution

Simplify Part (a)

Simplify Part (b)

Key Concepts

Multiplying Exponents

Power of a Power Rule

Simplifying Expressions with Exponents

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