Simplify (y³×x²)⁴: Solving Compound Variable Expressions

Power Rules with Compound Expressions

(y3×x2)4= (y^3\times x^2)^4=

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1

Understand the problem

(y3×x2)4= (y^3\times x^2)^4=

2

Step-by-step solution

We will solve the problem in two steps, in the first step we will use the power of a product rule:

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n The rule states that the power affecting a product within parentheses applies to each of the elements of the product when the parentheses are opened,

We begin by applying the law to the given problem:

(y3x2)4=(y3)4(x2)4 (y^3\cdot x^2)^4=(y^3)^4\cdot(x^2)^4 When we open the parentheses, we apply the power to each of the terms of the product separately, but since each of these terms is already raised to a power, we must be careful to use parentheses.

We then use the power of a power rule.

(bm)n=bmn (b^m)^n=b^{m\cdot n} We apply the rule to the given problem and we should obtain the following result:

(y3)4(x2)4=y34x24=y12x8 (y^3)^4\cdot(x^2)^4=y^{3\cdot4}\cdot x^{2\cdot4}=y^{12}\cdot x^8 When in the second step we perform the multiplication operation on the power exponents of the obtained terms.

Therefore, the correct answer is option d.

3

Final Answer

y12x8 y^{12}x^8

Key Points to Remember

Essential concepts to master this topic
  • Product Power Rule: Apply exponent to each factor separately
  • Technique: (y3x2)4=(y3)4(x2)4 (y^3 \cdot x^2)^4 = (y^3)^4 \cdot (x^2)^4
  • Check: Multiply exponents: 3×4=12 3 \times 4 = 12 and 2×4=8 2 \times 4 = 8

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of multiplying
    Don't add 3 + 4 = 7 to get y7x6 y^7x^6 ! This confuses the product rule with the power rule. Always multiply the exponents when raising a power to a power: (am)n=amn (a^m)^n = a^{m \cdot n} .

Practice Quiz

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

FAQ

Everything you need to know about this question

Why do I need to use parentheses around each factor?

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The parentheses ensure you apply the power of a power rule correctly! Without them, you might forget that y3 y^3 and x2 x^2 are already raised to powers.

How do I know when to multiply exponents versus add them?

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Multiply exponents when raising a power to a power: (am)n=amn (a^m)^n = a^{m \cdot n} . Add exponents when multiplying same bases: aman=am+n a^m \cdot a^n = a^{m+n} .

What if I get confused about which rule to use first?

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Always work from outside to inside! Start with the outermost exponent (the 4), then apply it to each factor inside the parentheses using the product power rule.

Can I simplify this expression differently?

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No, you must follow the order of operations. The exponent 4 applies to everything inside the parentheses, so you cannot rearrange or combine terms before applying the power rules.

Why is the answer y¹²x⁸ and not something simpler?

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This is the simplified form! Since y and x are different variables, you cannot combine them further. Each variable keeps its own exponent: y12 y^{12} and x8 x^8 .

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