times(a%5E(2))%5E(n%2B1))%2F((a%5E(4))%5E(-3))%3D(a%5E(3))%5E(3)--(a%5E(6))%5E(-3).jpeg "(a^(2n-3)times(a^(2))^(n+1))/((a^(4))^(-3))=(a^(3))^(3)- (a^(6))^(-3)")
Simplifying Exponential Expressions
This article explores the simplification of the following equation:
(a^(2n-3) * (a^(2))^(n+1)) / ((a^(4))^(-3)) = (a^(3))^(3) - (a^(6))^(-3)
We'll break down the steps to simplify each side of the equation and demonstrate their equivalence.
Simplifying the Left Hand Side
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Apply the power of a power rule: (a^m)^n = a^(m*n)
- (a^(2))^(n+1) = a^(2*(n+1)) = a^(2n+2)
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Apply the product of powers rule: a^m * a^n = a^(m+n)
- a^(2n-3) * a^(2n+2) = a^(2n-3 + 2n + 2) = a^(4n-1)
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Apply the power of a power rule again: (a^m)^n = a^(m*n)
- (a^(4))^(-3) = a^(4*(-3)) = a^(-12)
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Apply the quotient of powers rule: a^m / a^n = a^(m-n)
- a^(4n-1) / a^(-12) = a^(4n-1 - (-12)) = a^(4n + 11)
Therefore, the simplified left-hand side of the equation is a^(4n + 11).
Simplifying the Right Hand Side
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Apply the power of a power rule: (a^m)^n = a^(m*n)
- (a^(3))^(3) = a^(3*3) = a^9
- (a^(6))^(-3) = a^(6*(-3)) = a^(-18)
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Apply the difference of powers rule: a^m - a^n = a^n(a^(m-n) - 1)
- a^9 - a^(-18) = a^(-18)(a^(9 - (-18)) - 1) = a^(-18)(a^27 - 1)
Therefore, the simplified right-hand side of the equation is a^(-18)(a^27 - 1).
Comparing the Simplified Sides
We have shown that the left-hand side simplifies to a^(4n + 11) and the right-hand side simplifies to a^(-18)(a^27 - 1).
To prove the equivalence of these expressions, we need to find a value of 'n' that satisfies the equation. This requires further analysis and manipulation of the expressions.
It's important to note that this equation holds true only for specific values of 'n'.
This example demonstrates the importance of applying the rules of exponents to simplify complex expressions and reveals the potential for further manipulation and analysis to determine the validity of the equation.