(2^(2)y^(-6))/(8^(-1)z^(0)x^(-7))

Jasonbradley

3 min read

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Jun 16, 2024

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Simplifying Expressions with Exponents

In mathematics, simplifying expressions with exponents involves applying various rules to make the expression easier to understand and work with. Let's analyze the expression:

(2^(2)y^(-6))/(8^(-1)z^(0)x^(-7))

Understanding the Rules

Before diving into the simplification, let's review some crucial exponent rules:

• Product of powers: x^m * x^n = x^(m+n)
• Quotient of powers: x^m / x^n = x^(m-n)
• Power of a power: (x^m)^n = x^(m*n)
• Negative exponent: x^(-n) = 1/x^n
• Zero exponent: x^0 = 1

Simplifying the Expression

1. Rewrite 8 as 2^3: This helps us work with the same base. Our expression becomes: (2^(2)y^(-6))/((2^3)^(-1)z^(0)x^(-7))
2. Apply power of a power rule: (2^3)^(-1) = 2^(-3). The expression is now: (2^(2)y^(-6))/(2^(-3)z^(0)x^(-7))
3. Apply quotient of powers rule: This involves subtracting the exponents of the same base. The expression simplifies to: 2^(2-(-3))y^(-6)z^(0)x^(7)
4. Simplify the exponents: 2^(2-(-3)) = 2^(5). The expression now is: 2^(5)y^(-6)z^(0)x^(7)
5. Apply negative exponent and zero exponent rule: y^(-6) = 1/y^6 and z^0 = 1. The final simplified expression is: (32x^(7))/(y^(6))

Conclusion

By applying the rules of exponents systematically, we simplified the complex expression into a much more understandable form. This process demonstrates the power of these rules in simplifying mathematical expressions and making them easier to manipulate.