Simplify The Expression Completely: X − 5 Y − 8 = □ \frac{x^{-5}}{y^{-8}} = \square Y − 8 X − 5 ​ = □

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression x−5y−8\frac{x^{-5}}{y^{-8}} and asked to simplify it completely. To do this, we need to apply the rules of exponents, specifically the rule for dividing exponential expressions.

The Rule for Dividing Exponential Expressions

The rule for dividing exponential expressions states that when we divide two exponential expressions with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

where aa is the base and mm and nn are the exponents.

Applying the Rule to the Given Expression

Now that we have the rule, let's apply it to the given expression x−5y−8\frac{x^{-5}}{y^{-8}}. We can see that both expressions have a negative exponent, which means we need to apply the rule for dividing exponential expressions.

x−5y−8=y8x5\frac{x^{-5}}{y^{-8}} = \frac{y^{8}}{x^{5}}

Simplifying the Expression

Now that we have applied the rule, we can simplify the expression further by rewriting it in a more conventional form. We can do this by applying the rule for negative exponents, which states that:

a−n=1ana^{-n} = \frac{1}{a^n}

Using this rule, we can rewrite the expression as:

y8x5=1x5⋅y8\frac{y^{8}}{x^{5}} = \frac{1}{x^{5}} \cdot y^{8}

Final Simplification

Now that we have simplified the expression, we can see that it can be written in a more compact form. We can do this by combining the two terms into a single expression:

1x5⋅y8=y8x5\frac{1}{x^{5}} \cdot y^{8} = \frac{y^8}{x^5}

Conclusion

In this problem, we were given the expression x−5y−8\frac{x^{-5}}{y^{-8}} and asked to simplify it completely. We applied the rule for dividing exponential expressions and simplified the expression further by rewriting it in a more conventional form. Finally, we combined the two terms into a single expression, resulting in the simplified form y8x5\frac{y^8}{x^5}.

Key Takeaways

  • When dividing exponential expressions with the same base, we subtract the exponents.
  • When dealing with negative exponents, we can rewrite them as fractions using the rule a−n=1ana^{-n} = \frac{1}{a^n}.
  • We can simplify expressions by combining terms and rewriting them in a more conventional form.

Practice Problems

  • Simplify the expression x3y−2\frac{x^3}{y^{-2}}.
  • Simplify the expression a−4b2\frac{a^{-4}}{b^2}.
  • Simplify the expression c5d−3\frac{c^5}{d^{-3}}.

Solutions to Practice Problems

  • x3y−2=x3⋅y2=x3y2\frac{x^3}{y^{-2}} = x^3 \cdot y^2 = x^3y^2
  • a−4b2=1a4⋅1b2=1a4b2\frac{a^{-4}}{b^2} = \frac{1}{a^4} \cdot \frac{1}{b^2} = \frac{1}{a^4b^2}
  • c5d−3=c5⋅d3=c5d3\frac{c^5}{d^{-3}} = c^5 \cdot d^3 = c^5d^3

Real-World Applications

  • Exponents are used extensively in physics and engineering to describe the behavior of physical systems.
  • Exponents are used in finance to calculate interest rates and investment returns.
  • Exponents are used in computer science to describe the complexity of algorithms and data structures.

Conclusion

In this article, we simplified the expression x−5y−8\frac{x^{-5}}{y^{-8}} by applying the rule for dividing exponential expressions and rewriting it in a more conventional form. We also provided practice problems and solutions to help reinforce the concepts learned. Finally, we discussed the real-world applications of exponents and their importance in various fields.

Frequently Asked Questions

In this article, we'll address some of the most common questions related to simplifying exponential expressions. Whether you're a student struggling with algebra or a professional looking to brush up on your math skills, this Q&A section is designed to provide you with the answers you need.

Q: What is the rule for dividing exponential expressions?

A: The rule for dividing exponential expressions states that when we divide two exponential expressions with the same base, we subtract the exponents. Mathematically, this can be represented as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction using the rule a−n=1ana^{-n} = \frac{1}{a^n}. For example:

1x5=x−5\frac{1}{x^5} = x^{-5}

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent by applying the rule for dividing exponential expressions. For example:

x3y2=x3−2=x1=x\frac{x^3}{y^2} = x^{3-2} = x^1 = x

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

A: To simplify an expression with multiple bases, we need to apply the rule for dividing exponential expressions separately for each base. For example:

x3y2⋅z4w3=x3y2⋅z4w3=x3z4y2w3\frac{x^3}{y^2} \cdot \frac{z^4}{w^3} = \frac{x^3}{y^2} \cdot \frac{z^4}{w^3} = \frac{x^3z^4}{y^2w^3}

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the rule a0=1a^0 = 1. For example:

x3x3=x3−3=x0=1\frac{x^3}{x^3} = x^{3-3} = x^0 = 1

Q: How do I simplify an expression with a negative base?

A: To simplify an expression with a negative base, we need to apply the rule for negative exponents. For example:

1x5=x−5\frac{1}{x^5} = x^{-5}

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent by applying the rule for fractional exponents. For example:

x12=xx^{\frac{1}{2}} = \sqrt{x}

Q: How do I simplify an expression with a mixed exponent?

A: To simplify an expression with a mixed exponent, we need to apply the rule for dividing exponential expressions separately for each base. For example:

x3y2⋅z4w3=x3y2⋅z4w3=x3z4y2w3\frac{x^3}{y^2} \cdot \frac{z^4}{w^3} = \frac{x^3}{y^2} \cdot \frac{z^4}{w^3} = \frac{x^3z^4}{y^2w^3}

Conclusion

In this Q&A article, we've addressed some of the most common questions related to simplifying exponential expressions. Whether you're a student struggling with algebra or a professional looking to brush up on your math skills, this article is designed to provide you with the answers you need. By applying the rules for dividing exponential expressions, negative exponents, and fractional exponents, you can simplify even the most complex expressions and become a math master!