Reduce the Expression: 8^(3x) × 8^(3y) × 8^(2y+x)

Q: Why do I add the exponents instead of multiplying them?

The Product Rule states: $ a^m \times a^n = a^{m+n} $. When bases are the same (all 8's here), you add exponents. Multiplying exponents is for the Power Rule, which is different!

Q: How do I handle the parentheses in (2y+x)?

The parentheses show that (2y+x) is one complete exponent. Don't distribute or break it apart - just add it as a whole unit: $ 3x + 3y + (2y+x) $.

Q: What if the bases were different numbers?

If bases are different (like $ 8^2 \times 7^3 $), you cannot combine them using exponent rules. The product rule only works when bases are identical.

Q: How do I combine like terms in the exponent?

Group similar variables: collect all x terms (3x + x = 4x) and all y terms (3y + 2y = 5y). Then write as $ 4x + 5y $.

Q: Can I simplify this further than 8^(4x+5y)?

No, this is fully simplified! Since 4x and 5y have different variables, they cannot be combined further. $ 8^{4x+5y} $ is the final answer.

Exponent Rules with Multiple Terms

Reduce the following equation:

$8^{3x}\times8^{3y}\times8^{2y+x}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of exponents with equal bases (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:12 We will apply this formula to our exercise

00:16 We'll maintain the base and add the exponents together

00:36 Let's group the factors together

00:41 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$8^{3x}\times8^{3y}\times8^{2y+x}=$

Step-by-step solution

To solve this problem, let's simplify the expression $8^{3x} \times 8^{3y} \times 8^{2y+x}$ using exponent rules:

Step 1: Identify the exponents in each term:
- The first term is $8^{3x}$ with an exponent of $3x$ .
- The second term is $8^{3y}$ with an exponent of $3y$ .
- The third term is $8^{2y+x}$ with an exponent of $2y+x$ .

Step 2: Apply the multiplication of powers rule:
Since all terms have the same base of 8, add the exponents: $(3x) + (3y) + (2y + x)$ .

Step 3: Simplify the expression:
Adding the terms in the exponent gives us: $3x + 3y + 2y + x = 4x + 5y$ .

Therefore, the simplified expression is $8^{4x+5y}$ .

Final Answer

$8^{4x+5y}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying powers with same base, add exponents
Technique: Combine $3x + 3y + (2y+x) = 4x + 5y$
Check: Count all variable terms: 3x + x = 4x and 3y + 2y = 5y ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like $3x \times 3y \times (2y+x)$ = wrong complicated expression! This confuses the product rule with power rule. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The Product Rule states: $a^m \times a^n = a^{m+n}$ . When bases are the same (all 8's here), you add exponents. Multiplying exponents is for the Power Rule, which is different!

How do I handle the parentheses in (2y+x)?

The parentheses show that (2y+x) is one complete exponent. Don't distribute or break it apart - just add it as a whole unit: $3x + 3y + (2y+x)$ .

What if the bases were different numbers?

If bases are different (like $8^2 \times 7^3$ ), you cannot combine them using exponent rules. The product rule only works when bases are identical.

How do I combine like terms in the exponent?

Group similar variables: collect all x terms (3x + x = 4x) and all y terms (3y + 2y = 5y). Then write as $4x + 5y$ .

Can I simplify this further than 8^(4x+5y)?

No, this is fully simplified! Since 4x and 5y have different variables, they cannot be combined further. $8^{4x+5y}$ is the final answer.

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