Question

Which expression is equivalent to \(9^5 \cdot 9^3\) ? A. \(9^{15}\) B. \(81^8\) C. \(81^{15}\) D. \(9^8\)

Short answer

D. \(9^8\)

Step-by-step solution

Understand the Laws of Exponents

The problem involves multiplying powers with the same base. Use the rule: \[a^m \times a^n = a^{m+n}\]This rule states that when multiplying exponents with the same base, add their exponents.

Apply the Exponent Rule

Identify the base and exponents in the expression \(9^5 \cdot 9^3\). Here, the base is 9, and the exponents are 5 and 3. Using the exponent rule from Step 1, add the exponents together: \[5 + 3 = 8\]

Write the Equivalent Expression

Combine the base with the new exponent: \[9^5 \cdot 9^3 = 9^{8}\]

Choose the Correct Option

Look at the given choices and find the one that matches \(9^8\). The correct answer is D. \(9^8\).

Key concepts

Laws of Exponents

Exponents simplify repeated multiplication. With the laws of exponents, calculations become easier.
For example, one key law is:
\[a^m \times a^n = a^{m+n} \]
This means when multiplying two expressions with the same base, you add their exponents.
Another law is: \[\frac{a^m}{a^n} = a^{m-n} \]
This is useful when dividing expressions with the same base.
We also have: \[(a^m)^n = a^{m \times n} \]
This is essential for raising a power to another power.
These foundational rules are the basics you need to solve complex problems involving exponents.
Understanding the laws can help make difficult math simpler!

Multiplying Exponents

Multiplying exponents with the same base is straightforward.
Remember to add the exponents together!
Let's look at the problem: \(9^5 \times 9^3\).
Here, we have the same base (9) with different exponents (5 and 3).
Applying the rule, we add the exponents:
\(5 + 3 = 8\).
So, \(9^5 \times 9^3 = 9^8\).
This rule simplifies such problems effectively.
Now you know how to handle multiplying powers of the same base.

Equivalent Expressions

Finding equivalent expressions means transforming one expression into another that has the same value.
In our problem, the expression \(9^5 \times 9^3\) can be simplified using the laws of exponents.
We combined the exponents because the bases were the same, resulting in \(9^8\).
This process helps in identifying different ways to represent the same mathematical value.
In the options given: A. \(9^{15}\), B. \(81^8\), C. \(81^{15}\), D. \(9^8\), only D. \(9^8\) matches the simplified form.
Understanding equivalent expressions is vital, as it allows you to recognize and solve problems more efficiently!