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Chapter 10: Problem 6

$$\text { Fill in the blanks. } \quad a^{7} \cdot a=a \square^{+} \square=a^{8}$$

Short Answer

Expert verified
7

Step by step solution

01

Understand the problem

We need to fill in the blanks in the equation: \(a^{7} \times a = a^{\boxed{}} \times a = a^{8}\). The goal is to determine the exponent.
02

Apply the rules of exponents

Remember that when multiplying two expressions with the same base, you add the exponents. Therefore, \(a^m \times a^n = a^{m+n}\).
03

Find the missing exponent

Use the rule from step 2 to solve for the blank: \(a^{7} \times a = a^{7+1} = a^8\), so the blank should be filled with 7.
04

Confirm the solution

Check the equation with the filled blank: \(a^{7} \times a = a^8\) which confirms the calculation is correct.

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Key Concepts

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

Multiplying Exponents
When it comes to exponents, one of the fundamental operations you will encounter is multiplication. Multiplying exponents means combining two exponential terms with the same base by adding their exponents.

Consider the exercise: \(\). When you multiply exponential terms, you don't actually multiply the exponents themselves. Instead, you keep the base the same and add the exponents.

Here, the base is 'a', and the exponents are 7 and 1. When you multiply, you add these exponents to get \(a^{7+1} = a^8\). This is a direct application of the rule for multiplying exponents, helping you solve the problem easily.

Same Base Property
The same base property is crucial when working with exponents. This rule states that when you multiply exponential terms with the same base, you simply add the exponents. This property makes complex exponential operations more manageable.

In the equation given in the original exercise, we are working with the same base 'a' across both terms. This is represented as \(a^7 \times a^1 == a^{7+1} = a^8\).

By recognizing that the base remains the same, the solution becomes much more straightforward. Understanding and applying this property is key to mastering exponent multiplication effectively.

Adding Exponents
Adding exponents is the specific operation that occurs during the multiplication of terms with the same base. This rule simplifies potential confusion by stating exactly what should be done when bases match.

In our exercise, the problem simplifies to \(a^7 \times a = a^{8}\). When adding the exponents 7 and 1, we achieve the result of 8. The given bases (both 'a') remain consistent.

This rule can be applied to any exponential multiplication with the same base. For example, \(b^3 \times b^5 = b^{3+5} = b^8\). The process remains the same, making it a reliable method for solving similar problems.

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Most popular questions from this chapter

Simplify. $$\left(x^{60} y^{50}\right)^{2}$$

$$\text { Fill in the blanks. }\left(y^{2}\right)^{7}=y \square \cdot \square=y^{14}$$

$$\text { Fill in the blanks }\left(x^{3}\right)^{5}=x \square \cdot \square=x^{15}$$

Perform the indicated operations. $$\left(-5.1 y^{2}+4.8 y+2.3\right)+\left(-1.1 y^{2}-8.9 y+3.0\right)$$

Subtract the polynomials. $$\left(9 p^{3}-4 p^{2}+2 p\right)-\left(2 p^{3}+6 p^{2}-3 p\right)$$
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