Bounds: Powers and Roots (x ≥ 0)

Statement

When a positive quantity \(x\) has bounds, we can find the bounds for its powers and roots by applying the function to the interval endpoints. For \(x \geq 0\):

\[ L_{x^n} = (L_x)^n, \quad U_{x^n} = (U_x)^n \] \[ L_{\sqrt[n]{x}} = \sqrt[n]{L_x}, \quad U_{\sqrt[n]{x}} = \sqrt[n]{U_x} \]

Here \(L_x\) and \(U_x\) are the lower and upper bounds of \(x\). The rule works because powers and roots are increasing functions when \(x \geq 0\).

Why it’s true (short reason)

• If \(x\) is between two numbers, then raising it to a power or taking a root preserves that ordering, as long as \(x \geq 0\).
• For example, if \(L_x \leq x \leq U_x\), then \((L_x)^2 \leq x^2 \leq (U_x)^2\), and \(\sqrt{L_x} \leq \sqrt{x} \leq \sqrt{U_x}\).
• This works for any positive power \(n\) and any positive root, since these functions are monotonic (always increasing) over non-negative numbers.

Recipe (how to use it)

1. Find the interval for \(x\) from the rounding (e.g., nearest unit, d.p., or s.f.).
2. If you need \(x^n\):
   • Lower bound = \((L_x)^n\)
   • Upper bound = \((U_x)^n\)
3. If you need \(\sqrt[n]{x}\):
   • Lower bound = \(\sqrt[n]{L_x}\)
   • Upper bound = \(\sqrt[n]{U_x}\)
4. Give the interval clearly, with units if the context requires.

Spotting it

Look for this technique when:

• A number is rounded and then squared, cubed, or rooted in the question.
• Bounds are needed for areas, volumes, or scaling relationships.
• The exam asks for maximum and minimum values of a power or root.

Common pairings

• Squares and square roots in geometry (areas, diagonals).
• Cubes and cube roots in volumes.
• Fractional powers (e.g., \(x^{1/4}\)) for fourth roots.
• Bounds from rounding (nearest unit, d.p., s.f.).

Mini examples

1. \(x = 3\) (nearest 0.5). Then \(x \in [2.75, 3.25)\). Squaring: \(x^2 \in [2.75^2, 3.25^2] = [7.56, 10.56)\).
2. \(x = 5.4\) (1 d.p.). Then \(x \in [5.35, 5.45)\). Square root: \(\sqrt{x} \in [\sqrt{5.35}, \sqrt{5.45}) \approx [2.312, 2.335)\).
3. \(x \in [3.9, 4.1]\). Cube root: \(\sqrt[3]{x} \in [\sqrt[3]{3.9}, \sqrt[3]{4.1}) \approx [1.574, 1.611)\).

Pitfalls

• Forgetting positivity: These rules assume \(x \geq 0\). If negatives are possible, powers of odd/even exponents behave differently.
• Squaring incorrectly: Do not square the midpoint, always square the interval ends.
• Forgetting root monotonicity: Roots of positive numbers increase with the input, so always apply roots to both bounds directly.
• Mixing units: If the original \(x\) had units (e.g., metres), then \(x^2\) is in squared units (m\(^2\)), and cube roots return to the base units (m).

Exam strategy

• Write the rounding interval for \(x\) first, e.g., \(x=12\) (nearest integer) → \([11.5, 12.5)\).
• Apply the correct formula for power or root bounds.
• Keep calculations exact when possible, round at the end if required.
• Check whether units change (length → area → volume).
• In multiple choice, check which option matches the computed interval.

Extended micro-examples

1. Square bounds: \(x=6.2\) (1 d.p.) → \([6.15,6.25)\). Then \(x^2 \in [37.82, 39.06)\).
2. Cube bounds: \(x=2.0\) (2 s.f.) → \([1.95,2.05)\). Then \(x^3 \in [7.41,8.61)\).
3. Square root bounds: \(x=100\) (nearest 10) → \([95,105)\). Then \(\sqrt{x} \in [9.75,10.25)\).
4. Fourth root bounds: \(x \in [9.5,10.5)\). Then \(x^{1/4} \in [1.758,1.801)\).

Summary

For positive \(x\), powers and roots preserve the order of bounds. To find the range of \(x^n\) or \(\sqrt[n]{x}\), substitute the lower and upper bounds of \(x\) directly into the function. This method is especially useful in geometry and measurement problems, where rounded values are raised to powers (areas, volumes) or roots (side lengths). Always set up the interval first, apply the function carefully, and present the result with correct units.