Simplify: 8 X 3 Y − 2 Z − 4 6 X − 1 Y Z − 5 \frac{8 X^3 Y^{-2} Z^{-4}}{6 X^{-1} Y Z^{-5}} 6 X − 1 Y Z − 5 8 X 3 Y − 2 Z − 4 ​

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Introduction

When dealing with complex fractions involving exponents, it's essential to understand the rules of exponentiation and how to simplify expressions. In this article, we will focus on simplifying the given expression 8x3y−2z−46x−1yz−5\frac{8 x^3 y^{-2} z^{-4}}{6 x^{-1} y z^{-5}} using the rules of exponentiation.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, x3x^3 can be written as x⋅x⋅xx \cdot x \cdot x. When dealing with exponents, it's essential to understand the rules of exponentiation, including the product of powers rule, the power of a power rule, and the quotient of powers rule.

Product of Powers Rule

The product of powers rule states that when multiplying two powers with the same base, we add the exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Power of a Power Rule

The power of a power rule states that when raising a power to another power, we multiply the exponents. For example, (x2)3=x2⋅3=x6(x^2)^3 = x^{2 \cdot 3} = x^6.

Quotient of Powers Rule

The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents. For example, x2x3=x2−3=x−1\frac{x^2}{x^3} = x^{2-3} = x^{-1}.

Simplifying the Expression

To simplify the given expression 8x3y−2z−46x−1yz−5\frac{8 x^3 y^{-2} z^{-4}}{6 x^{-1} y z^{-5}}, we will apply the quotient of powers rule to the numerator and denominator separately.

Simplifying the Numerator

The numerator is 8x3y−2z−48 x^3 y^{-2} z^{-4}. We can rewrite this as 8⋅x3⋅y−2⋅z−48 \cdot x^3 \cdot y^{-2} \cdot z^{-4}.

Simplifying the Denominator

The denominator is 6x−1yz−56 x^{-1} y z^{-5}. We can rewrite this as 6⋅x−1⋅y⋅z−56 \cdot x^{-1} \cdot y \cdot z^{-5}.

Applying the Quotient of Powers Rule

Now that we have simplified the numerator and denominator, we can apply the quotient of powers rule to simplify the expression.

Simplifying the Expression

Using the quotient of powers rule, we can rewrite the expression as 8⋅x3⋅y−2⋅z−46⋅x−1⋅y⋅z−5=86⋅x3−(−1)⋅y−2−1⋅z−4−(−5)\frac{8 \cdot x^3 \cdot y^{-2} \cdot z^{-4}}{6 \cdot x^{-1} \cdot y \cdot z^{-5}} = \frac{8}{6} \cdot x^{3-(-1)} \cdot y^{-2-1} \cdot z^{-4-(-5)}.

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it by applying the rules of exponentiation.

Evaluating the Coefficient

The coefficient of the expression is 86=43\frac{8}{6} = \frac{4}{3}.

Evaluating the Exponents

The exponents of the expression are x3−(−1)=x4x^{3-(-1)} = x^4, y−2−1=y−3y^{-2-1} = y^{-3}, and z−4−(−5)=z1z^{-4-(-5)} = z^1.

Final Answer

The final answer is 43x4y−3z1\frac{4}{3} x^4 y^{-3} z^1.

Conclusion

In this article, we simplified the expression 8x3y−2z−46x−1yz−5\frac{8 x^3 y^{-2} z^{-4}}{6 x^{-1} y z^{-5}} using the rules of exponentiation. We applied the quotient of powers rule to simplify the expression and evaluated the resulting expression by applying the rules of exponentiation. The final answer is 43x4y−3z1\frac{4}{3} x^4 y^{-3} z^1.

Frequently Asked Questions

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, we add the exponents.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, we multiply the exponents.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, we subtract the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can apply the quotient of powers rule to simplify the expression and then evaluate the resulting expression by applying the rules of exponentiation.

Further Reading

Introduction

In our previous article, we simplified the expression 8x3y−2z−46x−1yz−5\frac{8 x^3 y^{-2} z^{-4}}{6 x^{-1} y z^{-5}} using the rules of exponentiation. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q&A

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

A: A power is the result of raising a number to a certain power, while an exponent is the number that is being raised to a certain power. For example, x2x^2 is a power, while 2 is the exponent.

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

A: To simplify an expression with negative exponents, you can rewrite the expression with positive exponents by moving the base to the other side of the fraction. For example, 1x−2=x2\frac{1}{x^{-2}} = x^2.

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

A: The rule for multiplying powers with the same base is to add the exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

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

A: The rule for dividing powers with the same base is to subtract the exponents. For example, x2x3=x2−3=x−1\frac{x^2}{x^3} = x^{2-3} = x^{-1}.

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

A: To simplify an expression with multiple bases, you can apply the rules of exponentiation separately to each base. For example, x2y3x4y2=x2−4y3−2=x−2y1=1x2y\frac{x^2 y^3}{x^4 y^2} = \frac{x^{2-4}}{y^{3-2}} = \frac{x^{-2}}{y^1} = \frac{1}{x^2 y}.

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

A: The rule for raising a power to another power is to multiply the exponents. For example, (x2)3=x2⋅3=x6(x^2)^3 = x^{2 \cdot 3} = x^6.

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

A: To simplify an expression with fractional exponents, you can rewrite the expression with integer exponents by multiplying the base by itself to the power of the denominator. For example, x12=xx^{\frac{1}{2}} = \sqrt{x}.

Tips and Tricks

  • When simplifying expressions with exponents, it's essential to apply the rules of exponentiation in the correct order.
  • When dealing with negative exponents, it's helpful to rewrite the expression with positive exponents by moving the base to the other side of the fraction.
  • When multiplying or dividing powers with the same base, it's essential to add or subtract the exponents, respectively.
  • When raising a power to another power, it's essential to multiply the exponents.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with exponents. We covered topics such as negative exponents, multiplying and dividing powers with the same base, and raising a power to another power. We also provided some tips and tricks for simplifying expressions with exponents.

Frequently Asked Questions

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

A: A power is the result of raising a number to a certain power, while an exponent is the number that is being raised to a certain power.

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

A: To simplify an expression with negative exponents, you can rewrite the expression with positive exponents by moving the base to the other side of the fraction.

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

A: The rule for multiplying powers with the same base is to add the exponents.

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

A: The rule for dividing powers with the same base is to subtract the exponents.

Further Reading