By Lisa Maloney | Updated Aug 30, 2022
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Exponents—symbols like y², x³, or the dreaded yx—can intimidate newcomers to algebra. In practice, removing them is often straightforward once you master a few basic techniques rooted in everyday arithmetic.
Sometimes exponent terms cancel themselves out. For example, consider:
\(y + 2x^2 – 5 = 2(x^2 + 2)\)
After expanding the right‑hand side you get:
\(y + 2x^2 – 5 = 2x^2 + 4\)
Notice that the \(2x^2\) terms are identical on both sides.
Subtract \(2x^2\) from each side, yielding
\(y – 5 = 4\)
Finally, add 5 to isolate y:
\(y = 9\)
While not every problem is this tidy, the strategy is a valuable first check.
Recognizing patterns that factor cleanly can eliminate exponents without solving step‑by‑step. Below are the most common formulas.
If the equation contains \(a^2 – b^2\), factor it as \((a + b)(a – b)\). For instance, \(x^2 – 16\) factors to \((x + 4)(x – 4)\).
When you see \(a^3 + b^3\), use \((a + b)(a^2 – ab + b^2)\). Example: \(y^3 + 8\) becomes \((y + 2)(y^2 – 2y + 4)\).
For \(a^3 – b^3\), the factorization is \((a – b)(a^2 + ab + b^2)\). Example: \(x^3 – 125\) factors to \((x – 5)(x^2 + 5x + 25)\).
Factoring often reduces the problem to simpler terms you can then solve or cancel in fractions.
When factoring is not applicable and you have a single exponent term, isolate it and then apply the corresponding root.
Example: \(z^3 – 25 = 2\). Add 25 to both sides to get \(z^3 = 27\).
Take the cube root of both sides: \(\sqrt[3]{z^3} = \sqrt[3]{27}\), simplifying to \(z = 3\).
