Choose The Correct Simplification Of $\left(2 X Y^2\right)^2\left(y^2\right)^3$.A. $2 X^2 Y^{10}$B. $4 X^2 Y^9$C. $4 X^2 Y^{10}$D. $2 X^2 Y^9$

by ADMIN 143 views

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the correct simplification of the expression (2xy2)2(y2)3\left(2 x y^2\right)^2\left(y^2\right)^3 and provide a step-by-step guide on how to simplify exponential expressions.

Understanding Exponents

Before we dive into the simplification of the given expression, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, in the expression x3x^3, the exponent 3 tells us to multiply the base number xx by itself three times: x3=xâ‹…xâ‹…xx^3 = x \cdot x \cdot x.

Simplifying the Given Expression

Now that we have a basic understanding of exponents, let's simplify the given expression (2xy2)2(y2)3\left(2 x y^2\right)^2\left(y^2\right)^3. To simplify this expression, we need to apply the rules of exponents.

Step 1: Apply the Power Rule

The power rule states that when we raise a power to another power, we multiply the exponents. In this case, we have (2xy2)2\left(2 x y^2\right)^2, which means we need to multiply the exponent 2 by the exponent 2. This gives us:

(2xy2)2=22x2y4\left(2 x y^2\right)^2 = 2^2 x^2 y^4

Step 2: Apply the Product Rule

The product rule states that when we multiply two powers with the same base, we add the exponents. In this case, we have 22x2y42^2 x^2 y^4 and (y2)3\left(y^2\right)^3, which means we need to add the exponents 4 and 3. This gives us:

22x2y4â‹…(y2)3=22x2y4+3=22x2y72^2 x^2 y^4 \cdot \left(y^2\right)^3 = 2^2 x^2 y^{4+3} = 2^2 x^2 y^7

Step 3: Simplify the Expression

Now that we have applied the power rule and the product rule, we can simplify the expression further. We can rewrite 222^2 as 4, which gives us:

22x2y7=4x2y72^2 x^2 y^7 = 4 x^2 y^7

Conclusion

In conclusion, the correct simplification of the expression (2xy2)2(y2)3\left(2 x y^2\right)^2\left(y^2\right)^3 is 4x2y74 x^2 y^7. We applied the power rule and the product rule to simplify the expression, and we were able to rewrite it in a simpler form.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. Here are a few:

  • Not applying the power rule: When raising a power to another power, we need to multiply the exponents. Failing to do so can lead to incorrect simplifications.
  • Not applying the product rule: When multiplying two powers with the same base, we need to add the exponents. Failing to do so can lead to incorrect simplifications.
  • Not simplifying the expression further: After applying the power rule and the product rule, we need to simplify the expression further by rewriting the exponents in a simpler form.

Tips for Simplifying Exponential Expressions

Here are a few tips for simplifying exponential expressions:

  • Read the expression carefully: Before simplifying the expression, read it carefully to make sure you understand what it says.
  • Apply the power rule first: When simplifying an expression, apply the power rule first to raise the power to another power.
  • Apply the product rule next: After applying the power rule, apply the product rule to multiply two powers with the same base.
  • Simplify the expression further: After applying the power rule and the product rule, simplify the expression further by rewriting the exponents in a simpler form.

Conclusion

Introduction

In our previous article, we explored the correct simplification of the expression (2xy2)2(y2)3\left(2 x y^2\right)^2\left(y^2\right)^3 and provided a step-by-step guide on how to simplify exponential expressions. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q&A

Q: What is the power rule in simplifying exponential expressions?

A: The power rule states that when we raise a power to another power, we multiply the exponents. For example, in the expression (x2)3\left(x^2\right)^3, the exponent 3 tells us to multiply the exponent 2 by itself three times: (x2)3=x2â‹…3=x6\left(x^2\right)^3 = x^{2 \cdot 3} = x^6.

Q: What is the product rule in simplifying exponential expressions?

A: The product rule states that when we multiply two powers with the same base, we add the exponents. For example, in the expression x2â‹…x3x^2 \cdot x^3, the exponents 2 and 3 tell us to add them together: x2â‹…x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you need to apply the power rule and the product rule in the correct order. First, apply the power rule to raise each power to another power. Then, apply the product rule to multiply two powers with the same base.

Q: What is the difference between a power and an exponent?

A: A power is the base number that is being raised to a certain power, while an exponent is the number that tells us how many times to multiply the base number by itself. For example, in the expression x3x^3, the base number is xx and the exponent is 3.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to rewrite the expression with positive exponents. For example, in the expression x−3x^{-3}, you can rewrite it as 1x3\frac{1}{x^3}.

Q: What is the rule for simplifying expressions with fractional exponents?

A: To simplify an expression with fractional exponents, you need to rewrite the expression with a whole number exponent. For example, in the expression x12x^{\frac{1}{2}}, you can rewrite it as x\sqrt{x}.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. Here are a few:

  • Not applying the power rule: When raising a power to another power, we need to multiply the exponents. Failing to do so can lead to incorrect simplifications.
  • Not applying the product rule: When multiplying two powers with the same base, we need to add the exponents. Failing to do so can lead to incorrect simplifications.
  • Not simplifying the expression further: After applying the power rule and the product rule, we need to simplify the expression further by rewriting the exponents in a simpler form.

Tips for Simplifying Exponential Expressions

Here are a few tips for simplifying exponential expressions:

  • Read the expression carefully: Before simplifying the expression, read it carefully to make sure you understand what it says.
  • Apply the power rule first: When simplifying an expression, apply the power rule first to raise the power to another power.
  • Apply the product rule next: After applying the power rule, apply the product rule to multiply two powers with the same base.
  • Simplify the expression further: After applying the power rule and the product rule, simplify the expression further by rewriting the exponents in a simpler form.

Conclusion

In conclusion, simplifying exponential expressions is an essential skill for students and professionals alike. By applying the power rule and the product rule, we can simplify expressions and rewrite them in a simpler form. Remember to read the expression carefully, apply the power rule first, apply the product rule next, and simplify the expression further to get the correct simplification.