- Use the rule \(a^{-n} = \frac{1}{a^n}\) to rewrite terms and simplify the given equations.
- Equate the powers on both sides to solve for the unknown variables.
- The calculated values are \(x = -5\) and \(y = 4\).
- Final Answer: \(\boxed{x = -5, y = 4}\).
Explanation
- Determining the value of x
To find the value of \(x\) in the equation \(g^x + h^{-3} = \frac{1}{g^5} + \frac{1}{h^3}\), we utilize the exponent rule \(a^{-n} = \frac{1}{a^n}\). Applying this rule to the term \(h^{-3}\), we get \(h^{-3} = \frac{1}{h^3}\). Substituting this into our equation yields \(g^x + \frac{1}{h^3} = \frac{1}{g^5} + \frac{1}{h^3}\). By subtracting \(\frac{1}{h^3}\) from both sides, the equation simplifies to \(g^x = \frac{1}{g^5}\). Again, using the rule \(a^{-n} = \frac{1}{a^n}\), we rewrite \(\frac{1}{g^5}\) as \(g^{-5}\). Thus, we have \(g^x = g^{-5}\), which implies that \(x = -5\).
- Determining the value of y
To find the value of \(y\) in the equation \(b^4 c^{-7} = \frac{b^y}{c^7}\), we again utilize the exponent rule \(a^{-n} = \frac{1}{a^n}\). Applying this rule to the term \(c^{-7}\) on the left side, we get \(c^{-7} = \frac{1}{c^7}\). Our equation becomes \(b^4 \cdot \frac{1}{c^7} = \frac{b^y}{c^7}\), which simplifies to \(\frac{b^4}{c^7} = \frac{b^y}{c^7}\). By multiplying both sides of the equation by \(c^7\) (assuming \(c
eq 0\)), we obtain \(b^4 = b^y\). Comparing the exponents of base \(b\), it follows that \(y = 4\).
- Final Conclusion
Through the application of fundamental exponent laws, we have determined that \(x = -5\) and \(y = 4\). The values are \(\boxed{x = -5}\) and \(\boxed{y = 4}\).
Examples
Exponent laws are fundamental tools in mathematics used to simplify complex algebraic expressions efficiently. Imagine you are working with large data sets in computer science where memory consumption is measured in powers of bytes (like KB, MB, GB). Being able to manipulate and equate these expressions using exponent rules allows for quick calculations of storage requirements and data transfer speeds, ensuring that systems remain optimized and balanced. Similarly, in fields like physics and engineering, these laws help standardize units and scales, enabling more accurate measurements and predictions.