Simplify: $\sqrt{x^5 \cdot x^{-6}}$

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03-11-2026
Mathematics

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Simplify: $\sqrt{x^5 \cdot x^{-6}}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T12:37:08-05:00

Use the exponent rule $x^a \cdot x^b = x^{a+b}$ to simplify the radicand $x^5 \cdot x^{-6}$ to $x^{-1}$.
Express the square root of $x^{-1}$ as $\sqrt{\frac{1}{x}}$, which simplifies to $\frac{1}{\sqrt{x}}$ for $x > 0$.
Rationalize the denominator by multiplying the expression by $\frac{\sqrt{x}}{\sqrt{x}}$ to arrive at the final simplified form: $\boxed{\frac{\sqrt{x}}{x}}$.

Explanation

Simplifying the Radicand

To simplify the expression $\sqrt{x^5 \cdot x^{-6}}$, we first examine the radicand, which is the part inside the square root. According to the rules of exponents, when multiplying two powers with the same base, we add their exponents. Specifically, for any non-zero real number $x$, we have $x^a \cdot x^b = x^{a+b}$. In our case, this gives:
\[x^5 \cdot x^{-6} = x^{5 + (-6)} = x^{-1}\]
Thus, the expression can be rewritten as $\sqrt{x^{-1}}$.

Applying the Square Root Property

Now that we have the expression in the form $\sqrt{x^{-1}}$, we can rewrite $x^{-1}$ as $\frac{1}{x}$. So, the expression becomes $\sqrt{\frac{1}{x}}$.
For this expression to be defined as a real number, the value under the square root must be non-negative. Therefore, we require $\frac{1}{x} > 0$, which implies that $x$ must be a positive real number ($x > 0$). Under this condition, we can simplify the square root of the fraction by taking the square root of the numerator and the denominator separately:
\[\sqrt{\frac{1}{x}} = \frac{\sqrt{1}}{\sqrt{x}} = \frac{1}{\sqrt{x}}\]

Rationalizing the Denominator and Final Result

To express the result in a more standard form, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{x}$:
\[\frac{1}{\sqrt{x}} = \frac{1 \cdot \sqrt{x}}{\sqrt{x} \cdot \sqrt{x}} = \frac{\sqrt{x}}{x}\]
Thus, the expression $\sqrt{x^5 \cdot x^{-6}}$ simplifies to $\frac{\sqrt{x}}{x}$ for $x > 0$. The final answer is $\boxed{\frac{\sqrt{x}}{x}}$.

Examples

Simplifying algebraic expressions like $\sqrt{x^5 \cdot x^{-6}}$ is a fundamental skill in algebra, useful in fields ranging from physics to engineering. For example, in calculating the intensity of light or sound as it spreads from a source, formulas often involve inverse square relationships where exponents must be combined and simplified. Mastering these exponent rules allows you to handle complex formulas efficiently, making it easier to solve problems involving rates, scaling, or physical laws.