Simplify the Expression: 6^x × 6^a × 6^b Using Exponent Rules

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

The exponent multiplication rule states: when bases are the same, add the exponents. Think of it as $ 6^x \times 6^a = 6 \times 6 \times ... $ (x times) × $ 6 \times 6 \times ... $ (a times) = $ 6^{x+a} $!

Q: What if the bases were different numbers?

If the bases are different (like $ 6^x \times 5^a $), you cannot combine them using exponent rules. The bases must be exactly the same to add exponents.

Q: How can I remember not to multiply the exponents?

Remember: multiplication of same bases → addition of exponents. Only when raising a power to another power do you multiply exponents, like $ (6^x)^a = 6^{xa} $.

Exponent Multiplication with Multiple Variables

Reduce the following equation

$6^x\times6^a\times6^b=$

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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 powers with equal bases (A)

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

00:11 We will apply this formula to our exercise

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

00:25 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

$6^x\times6^a\times6^b=$

Step-by-step solution

To solve this problem, we'll simplify the given equation using the rules of exponents:

Identify that all terms in the product $6^x \times 6^a \times 6^b$ have the same base, which is 6.
Apply the exponent multiplication rule: When multiplying powers with the same base, we add the exponents together. Therefore, the expression becomes $6^{x+a+b}$ .

By applying this exponent rule, we determine that the simplified expression is $6^{x+a+b}$ .

Therefore, the solution to the problem is $6^{x+a+b}$ .

Final Answer

$6^{x+a+b}$

Key Points to Remember

Essential concepts to master this topic

Same Base Rule: When multiplying powers with identical bases, add the exponents
Technique: $6^x \times 6^a \times 6^b = 6^{x+a+b}$ by adding x+a+b
Check: Verify base stays 6 and exponents are added, not multiplied ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like x×a×b = 6^{abx}! This confuses the multiplication rule with the power rule. When multiplying same bases, the exponents add together. Always add exponents: x+a+b 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 we add the exponents instead of multiplying them?

The exponent multiplication rule states: when bases are the same, add the exponents. Think of it as $6^x \times 6^a = 6 \times 6 \times ...$ (x times) × $6 \times 6 \times ...$ (a times) = $6^{x+a}$ !

What if the bases were different numbers?

If the bases are different (like $6^x \times 5^a$ ), you cannot combine them using exponent rules. The bases must be exactly the same to add exponents.

Does it matter how many terms I'm multiplying?

No! Whether you have 2, 3, or more terms with the same base, just add all the exponents together. The rule works for any number of terms: $6^a \times 6^b \times 6^c \times 6^d = 6^{a+b+c+d}$ .

How can I remember not to multiply the exponents?

Remember: multiplication of same bases → addition of exponents. Only when raising a power to another power do you multiply exponents, like $(6^x)^a = 6^{xa}$ .

What if some exponents are negative?

The rule still applies! Add all exponents together, including negative ones. For example: $6^3 \times 6^{-1} \times 6^x = 6^{3+(-1)+x} = 6^{2+x}$ .

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