Expand a^(3+5): Simplifying Variable Exponent Expression

Q: Why does $ a^{3+5} $ become multiplication instead of addition?

The exponent rule states that when you have the same base with exponents that add, you separate them by multiplying: $ a^{m+n} = a^m \times a^n $. This is different from adding the expressions themselves!

Q: Can I just calculate 3+5=8 and write $ a^8 $ ?

While $ a^{3+5} = a^8 $ is mathematically correct, the question asks you to expand the expression. This means showing it as $ a^3 \times a^5 $ to demonstrate your understanding of exponent rules.

Q: What's the difference between $ a^3 \times a^5 $ and $ a^3 + a^5 $ ?

$ a^3 \times a^5 $ means you're multiplying two powers of the same base, which equals $ a^8 $. But $ a^3 + a^5 $ means you're adding two different expressions, which cannot be simplified further.

Q: How do I remember this exponent rule?

Think of it this way: $ a^3 $ means three a's multiplied together, and $ a^5 $ means five a's multiplied together. When you multiply them, you get eight a's total: $ a^8 $!

Q: Does this rule work with any numbers in the exponent?

Yes! The rule $ a^{m+n} = a^m \times a^n $ works with any real numbers for m and n, including fractions, negatives, and decimals. The base just needs to stay the same.

Exponent Rules with Addition

Expand the following equation:

$a^{3+5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Identify which expressions are equal to the original expression

00:03 According to the laws of exponents, multiplying powers with the same base (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:11 We'll apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:18 We can observe that this expression equals the original expression

00:21 We'll use the same method in order to simplify the remaining expressions

00:24 In this expression, the operation is addition and not multiplication, therefore it's not relevant

00:33 Any number raised to the power of 1 always equals itself

00:36 We'll apply this formula to our exercise, and raise to the power of 1

00:46 This expression is not equal to the original expression

00:51 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

$a^{3+5}=$

Step-by-step solution

To solve this problem, we begin by rewriting the expression that incorporates exponent rules. The expression given is $a^{3+5}$ . According to the rule of exponents, when you have a base raised to a power that is a sum, $a^{m+n} = a^m \times a^n$ .

Let's apply this rule:

Write the exponent as a sum: $3 + 5$ .
Apply the exponent rule: $a^{3+5}$ becomes $a^3 \times a^5$ .

Thus, the expanded form of $a^{3+5}$ using the rule of exponents is $a^3 \times a^5$ .

Finally, comparing with the provided options, choice 1 ( $a^3 \times a^5$ ) is the correct one, as it correctly uses the exponent rule.

Therefore, the solution to the problem is $a^3\times a^5$ .

Final Answer

$a^3\times a^5$

Key Points to Remember

Essential concepts to master this topic

Rule: When exponents are added in power: $a^{m+n} = a^m \times a^n$
Technique: Split $a^{3+5}$ into $a^3 \times a^5$ using exponent rule
Check: Verify that $a^8 = a^3 \times a^5$ by counting total powers ✓

Common Mistakes

Avoid these frequent errors

Adding bases instead of multiplying them
Don't write $a^{3+5} = a^3 + a^5$ = completely different expression! This treats addition like exponent addition, but you're actually working with multiplication of bases. Always use $a^{m+n} = a^m \times a^n$ when expanding exponents.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does $a^{3+5}$ become multiplication instead of addition?

The exponent rule states that when you have the same base with exponents that add, you separate them by multiplying: $a^{m+n} = a^m \times a^n$ . This is different from adding the expressions themselves!

Can I just calculate 3+5=8 and write $a^8$ ?

While $a^{3+5} = a^8$ is mathematically correct, the question asks you to expand the expression. This means showing it as $a^3 \times a^5$ to demonstrate your understanding of exponent rules.

What's the difference between $a^3 \times a^5$ and $a^3 + a^5$ ?

$a^3 \times a^5$ means you're multiplying two powers of the same base, which equals $a^8$ . But $a^3 + a^5$ means you're adding two different expressions, which cannot be simplified further.

How do I remember this exponent rule?

Think of it this way: $a^3$ means three a's multiplied together, and $a^5$ means five a's multiplied together. When you multiply them, you get eight a's total: $a^8$ !

Does this rule work with any numbers in the exponent?

Yes! The rule $a^{m+n} = a^m \times a^n$ works with any real numbers for m and n, including fractions, negatives, and decimals. The base just needs to stay the same.

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Expand a^(3+5): Simplifying Variable Exponent Expression

Exponent Rules with Addition

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does $a^{3+5}$ become multiplication instead of addition?

Can I just calculate 3+5=8 and write $a^8$ ?

What's the difference between $a^3 \times a^5$ and $a^3 + a^5$ ?

How do I remember this exponent rule?

Does this rule work with any numbers in the exponent?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: 4^(a+b+c) Using Power Properties

Expand the Expression: Breaking Down g^(10a+5x)

Expand t^9: Writing Out the Ninth Power Expression

Expand b^7: Calculating the Seventh Power Expression

Expand a^(3+5): Simplifying Variable Exponent Expression

Exponent Rules with Addition

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does a3+5 a^{3+5} a3+5 become multiplication instead of addition?

Can I just calculate 3+5=8 and write a8 a^8 a8?

What's the difference between a3×a5 a^3 \times a^5 a3×a5 and a3+a5 a^3 + a^5 a3+a5?

How do I remember this exponent rule?

Does this rule work with any numbers in the exponent?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: 4^(a+b+c) Using Power Properties

Expand the Expression: Breaking Down g^(10a+5x)

Expand t^9: Writing Out the Ninth Power Expression

Expand b^7: Calculating the Seventh Power Expression

Why does $a^{3+5}$ become multiplication instead of addition?

Can I just calculate 3+5=8 and write $a^8$ ?

What's the difference between $a^3 \times a^5$ and $a^3 + a^5$ ?