Simplify The Following By Using Exponential Laws. Leave The Answer In Positive Exponents And Surd Form If Applicable.${ \left(-2x \sqrt[3]{x^6 Y {12}}\right)\left(3x 5 Y^6\right) X {-5}\left(y 5\right) 0\left(-2x 3\right)^2 }$

Introduction

In this article, we will simplify the given expression using exponential laws. The expression involves various mathematical operations, including multiplication, exponentiation, and surds. We will apply the laws of exponents to simplify the expression and leave the answer in positive exponents and surd form if applicable.

The Given Expression

The given expression is:

${ \left(-2x \sqrt[3]{x^6 y^{12}}\right)\left(3x^5 y^6\right) x^{-5}\left(y^5\right)^0\left(-2x^3\right)^2 }$

Step 1: Simplify the Expression Inside the Parentheses

We will start by simplifying the expression inside the parentheses. The expression inside the parentheses is:

${ -2x \sqrt[3]{x^6 y^{12}} }$

We can rewrite the expression as:

${ -2x \cdot x^{\frac{6}{3}} \cdot y^{\frac{12}{3}} }$

Using the laws of exponents, we can simplify the expression as:

${ -2x \cdot x^2 \cdot y^4 }$

${ = -2x^3 y^4 }$

Step 2: Simplify the Expression Outside the Parentheses

Next, we will simplify the expression outside the parentheses. The expression outside the parentheses is:

${ 3x^5 y^6 }$

This expression is already simplified.

Step 3: Simplify the Expression with the Negative Exponent

The expression with the negative exponent is:

${ x^{-5} }$

Using the laws of exponents, we can rewrite the expression as:

${ \frac{1}{x^5} }$

Step 4: Simplify the Expression with the Zero Exponent

The expression with the zero exponent is:

${ \left(y^5\right)^0 }$

Using the laws of exponents, we can simplify the expression as:

${ 1 }$

Step 5: Simplify the Expression with the Exponent 2

The expression with the exponent 2 is:

${ \left(-2x^3\right)^2 }$

Using the laws of exponents, we can rewrite the expression as:

${ (-2)^2 \cdot (x^3)^2 }$

${ = 4 \cdot x^6 }$

Step 6: Multiply the Simplified Expressions

Now, we will multiply the simplified expressions:

${ -2x^3 y^4 \cdot 3x^5 y^6 \cdot \frac{1}{x^5} \cdot 1 \cdot 4x^6 }$

Using the laws of exponents, we can simplify the expression as:

${ -6x^3 y^4 \cdot x^5 \cdot x^6 }$

${ = -6x^14 y^4 }$

Conclusion

In this article, we simplified the given expression using exponential laws. We applied the laws of exponents to simplify the expression and left the answer in positive exponents and surd form if applicable. The final simplified expression is:

${ -6x^14 y^4 }$

Key Takeaways

• The laws of exponents can be used to simplify expressions involving exponents and surds.
• The expression inside the parentheses can be simplified using the laws of exponents.
• The expression outside the parentheses can be simplified using the laws of exponents.
• The expression with the negative exponent can be rewritten using the laws of exponents.
• The expression with the zero exponent can be simplified using the laws of exponents.
• The expression with the exponent 2 can be rewritten using the laws of exponents.
• The simplified expressions can be multiplied using the laws of exponents.

Final Answer

The final simplified expression is:

${ -6x^14 y^4 }$

Q: What are exponential laws?

A: Exponential laws are mathematical rules that govern the behavior of exponents. They provide a way to simplify expressions involving exponents and surds.

Q: How do I apply exponential laws to simplify expressions?

A: To apply exponential laws, you need to follow these steps:

1. Simplify the expression inside the parentheses using the laws of exponents.
2. Simplify the expression outside the parentheses using the laws of exponents.
3. Rewrite the expression with negative exponents using the laws of exponents.
4. Simplify the expression with zero exponents using the laws of exponents.
5. Rewrite the expression with exponents using the laws of exponents.
6. Multiply the simplified expressions using the laws of exponents.

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base is:

${ a^m \cdot a^n = a^{m+n} }$

Q: What is the rule for dividing exponents with the same base?

A: The rule for dividing exponents with the same base is:

${ \frac{a^m}{a^n} = a^{m-n} }$

Q: What is the rule for raising a power to a power?

A: The rule for raising a power to a power is:

${ (a^m)^n = a^{m \cdot n} }$

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

A: The rule for simplifying expressions with negative exponents is:

${ a^{-m} = \frac{1}{a^m} }$

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

A: The rule for simplifying expressions with zero exponents is:

${ a^0 = 1 }$

Q: How do I simplify expressions involving surds?

A: To simplify expressions involving surds, you need to follow these steps:

1. Simplify the expression inside the parentheses using the laws of exponents.
2. Simplify the expression outside the parentheses using the laws of exponents.
3. Rewrite the expression with surds using the laws of exponents.
4. Multiply the simplified expressions using the laws of exponents.

Q: What are some common mistakes to avoid when simplifying expressions using exponential laws?

A: Some common mistakes to avoid when simplifying expressions using exponential laws include:

• Not following the order of operations (PEMDAS)
• Not simplifying expressions inside parentheses
• Not rewriting expressions with negative exponents
• Not simplifying expressions with zero exponents
• Not multiplying simplified expressions using the laws of exponents

Q: How do I check my work when simplifying expressions using exponential laws?

A: To check your work when simplifying expressions using exponential laws, you need to follow these steps:

1. Simplify the expression using the laws of exponents.
2. Check if the simplified expression is in the correct form.
3. Verify that the simplified expression is equivalent to the original expression.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions using exponential laws. We covered topics such as the rules for multiplying and dividing exponents, raising a power to a power, and simplifying expressions with negative and zero exponents. We also discussed common mistakes to avoid and how to check your work when simplifying expressions using exponential laws.