Simplify Powers of 8: Combining 8^(-3x), 8^(-3y), and 8^(2y+x)

Q: Why do we add exponents instead of multiplying them?

The multiplication rule for exponents states that $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 8^2 \times 8^3 = (8 \times 8) \times (8 \times 8 \times 8) = 8^5 $, which is $ 8^{2+3} $!

Q: What if the variables are mixed together like 2y+x?

Keep the entire expression as one unit! The exponent $ (2y+x) $ stays together, so you add it as a whole: $ (-3x) + (-3y) + (2y+x) $.

Q: Can I simplify the variables inside the exponent?

Yes! After adding all exponents, combine like terms: $ -3x + x = -2x $ and $ -3y + 2y = -y $, giving you $ 8^{-2x-y} $.

Exponent Rules with Multiple Variable 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

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

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

00:11 We'll apply this formula to our exercise, one operation at a time

00:14 We'll maintain the base and add together the exponents

00:32 We'll group the factors together

00:43 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, we need to simplify the expression $8^{-3x} \times 8^{-3y} \times 8^{2y+x}$ using exponent rules. Let's break it down step-by-step.

First, recognize that each term in the product has the same base, which is 8. The multiplication rule for exponents allows us to add the exponents when multiplying like bases. Our expression is:

$8^{-3x} \times 8^{-3y} \times 8^{2y+x}$

According to the rule $a^m \times a^n = a^{m+n}$ , we add the exponents of each term:

$(-3x) + (-3y) + (2y + x)$

Combine the exponents inside the parentheses:

$-3x - 3y + 2y + x$

Now, group and simplify like terms:

The terms with $x$ : $-3x + x = -2x$
The terms with $y$ : $-3y + 2y = -y$

Combine these results:

The expression becomes $-2x - y$ .

Therefore, the reduced form of the original expression is:

$8^{-2x-y}$

Final Answer

$8^{-2x-y}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add all the exponents together
Technique: Group like variables: (-3x + x) = -2x, (-3y + 2y) = -y
Check: Verify by expanding: $8^{-3x} \times 8^{-3y} \times 8^{2y+x} = 8^{(-3x) + (-3y) + (2y+x)}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like (-3x) × (-3y) × (2y+x) = complicated mess! This gives completely wrong results because you're using the wrong operation. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

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Simplify the following equation:

$5^8\times5^3=$

FAQ

Everything you need to know about this question

Why do we add exponents instead of multiplying them?

The multiplication rule for exponents states that $a^m \times a^n = a^{m+n}$ . Think of it this way: $8^2 \times 8^3 = (8 \times 8) \times (8 \times 8 \times 8) = 8^5$ , which is $8^{2+3}$ !

How do I handle negative exponents in the addition?

Treat negative exponents just like negative numbers in regular addition. For example: $(-3x) + (-3y) = -3x - 3y$ . Don't forget to keep the negative signs when combining!

What if the variables are mixed together like 2y+x?

Keep the entire expression as one unit! The exponent $(2y+x)$ stays together, so you add it as a whole: $(-3x) + (-3y) + (2y+x)$ .

Can I simplify the variables inside the exponent?

Yes! After adding all exponents, combine like terms: $-3x + x = -2x$ and $-3y + 2y = -y$ , giving you $8^{-2x-y}$ .

What if I get confused with all the variables?

Write out each step clearly! First: identify all exponents. Second: add them together. Third: group like terms (all x's together, all y's together). Take your time - accuracy beats speed!

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