Question

Simplify \(7^3 \cdot 7^2 \cdot 7^5\). A. \(7^{30}\) B. \(7^{10}\) C. \(343^{30}\) D. \(343^{10}\)

Short answer

Option B: \(7^{10}\).

Step-by-step solution

Understand the Problem

The problem requires simplifying the expression involving exponentiation: \(7^3 \cdot 7^2 \cdot 7^5\). The goal is to express the product as a single exponent.

Recall the Product of Powers Property

When multiplying powers with the same base, add their exponents. This is given by the rule: \(a^m \cdot a^n = a^{m+n}\). Here, all bases are 7.

Apply the Property to the Given Expression

Add the exponents together: \(3 + 2 + 5\). This is calculated as follows:\(7^3 \cdot 7^2 \cdot 7^5 = 7^{(3+2+5)}\)

Calculate the Sum of Exponents

Calculate the sum: \(3 + 2 + 5 = 10\). Therefore, the expression simplifies to \(7^{10}\).

Compare Options to Find the Correct Answer

The simplified expression, \(7^{10}\), matches option B.

Key concepts

product of powers property

When working with exponents, a crucial property to understand is the **Product of Powers Property**. This property states that when you multiply two expressions that have the same base, you can add the exponents. Mathematically, it's expressed as: \[ a^m \times a^n = a^{m+n} \] In this exercise, the base is 7 for all the terms: \[ 7^3 \times 7^2 \times 7^5 \] Since the base is the same (7), we add the exponents together: \[ 3 + 2 + 5 \] This will help us convert the expression into a simpler form with a single exponent. Understanding this property is essential when dealing with exponentiation and will make simplifying such expressions much easier.

simplifying expressions

Simplifying expressions makes them more manageable and easier to understand. Start by identifying any applicable rules or properties, like the **Product of Powers Property**. Let's use this property to simplify:\[ 7^3 \times 7^2 \times 7^5 \] First, we recognize that all terms have the same base, which is 7. Following the property, we add the exponents together:\[ 3 + 2 + 5 \] Which equals 10. Therefore, the expression simplifies to:\[ 7^{10} \] Simplifying allows us to write complicated expressions in a more concise and easier to interpret way. This skill will not only help with this specific problem but is also applicable across various math problems involving exponents.

adding exponents

In expressions with exponents, **adding exponents** is a straightforward step when you have the same base. Look at the given expression: \[ 7^3 \times 7^2 \times 7^5 \] All parts of this expression share the same base of 7. This means we can apply the Product of Powers Property and add the exponents together: \[ 3 + 2 + 5 \] To add exponents:

• Identify the base (the number being raised to a power).
• Add the numbers in the exponents.

The sum of 3, 2, and 5 is 10, so the simplified expression is: \[ 7^{10} \] Regularly practice this method, and soon you'll find adding exponents to be intuitive and easy.