Solve Nested Exponents: ((14^(3x))^(2y))^(5a) Simplification

Power Rules with Multiple Nested Exponents

((143x)2y)5a= ((14^{3x})^{2y})^{5a}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:10 Let's simplify this expression together.
00:14 When you have a power raised to another power, multiply the exponents together.
00:19 Now, let's use this rule in our example.
00:23 Multiply each of the exponents carefully.
00:31 Take it step by step, solving one multiplication at a time.
00:45 Great job! That's your solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

((143x)2y)5a= ((14^{3x})^{2y})^{5a}=

2

Step-by-step solution

Using the power rule for an exponent raised to another exponent:

(am)n=amn (a^m)^n=a^{m\cdot n} We apply the rule to the given problem:

((143x)2y)5a=(143x)2y5a=143x2y5a=1430xya ((14^{3x})^{2y})^{5a}=(14^{3x})^{2y\cdot5a}=14^{3x\cdot2y\cdot5a}=14^{30xya} In the first step we applied the aforementioned power rule and removed the outer parentheses. In the next step we again applied the power rule and removed the remaining parentheses.

In the final step we simplified the resulting expression,

Therefore, through the rule of substitution (which is applied to the exponent of the power in the obtained expression) it can be concluded that the correct answer is answer D.

3

Final Answer

1430axy 14^{30axy}

Key Points to Remember

Essential concepts to master this topic
  • Power Rule: When raising a power to a power, multiply the exponents
  • Technique: Apply rule step by step: (143x)2y=143x2y=146xy (14^{3x})^{2y} = 14^{3x \cdot 2y} = 14^{6xy}
  • Check: Count total multiplications in exponent: 3x × 2y × 5a = 30axy ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of multiplying
    Don't add exponents like 3x + 2y + 5a = wrong answer! This completely ignores the power rule and gives incorrect results. Always multiply the exponents when raising a power to another power.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

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The power rule says (am)n=amn (a^m)^n = a^{m \cdot n} . When you raise a power to another power, you're essentially multiplying the base by itself m times, n times, which means m × n total multiplications!

How do I handle multiple layers of parentheses?

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Work from the outside in, applying the power rule one step at a time. First remove the outermost parentheses, then the next layer, until you have a single exponent.

What's the difference between this and the product rule?

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The product rule is for multiplying powers with the same base: aman=am+n a^m \cdot a^n = a^{m+n} (you ADD). The power rule is for raising a power to a power: (am)n=amn (a^m)^n = a^{m \cdot n} (you MULTIPLY).

Can I solve this problem in a different order?

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Yes! You can work from inside out or outside in. Both methods give the same answer: 1430axy 14^{30axy} . Choose the approach that feels most comfortable to you.

How do I remember when to multiply vs add exponents?

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Memory trick: Parentheses around powers = multiply exponents. Powers being multiplied together = add exponents. Look for those parentheses as your clue!

What if the variables were in a different order?

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Order doesn't matter in multiplication! Whether you get 1430axy 14^{30axy} , 1430xya 14^{30xya} , or 1430yax 14^{30yax} , they're all the same answer.

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