1.5 Exponents, roots, and rational exponents

Repeated multiplication and its inverse operations provide compact ways to express area, volume, scaling, and scientific notation.

Exponent. A number that indicates the power to which a base is raised.

For example, \[2^5 = 32.\] The quantity 5 in \(2^5\) is the exponent.

For real numbers \(a\), \(b\), \(m\) and \(n\), we have

\[a^ma^n=a^{m+n},\qquad \qquad \frac{a^m}{a^n}=a^{m-n},\qquad \qquad (a^m)^n=a^{mn}.\]

Worked Example: Simplifying exponent notation

\[\frac{x^7x^{-2}}{x^3}=x^{7-2-3}=x^2,\qquad x\ne0.\]

Try It. Simplify \((a^3)^2/a^4\).
Check Your Work \(a^2\), with \(a\ne0\).

Scientific notation expresses a number in the form \(a\times10^n\), where \(1\leq |a|<10\) and \(n\) is an integer. It is especially useful in water engineering for representing very large or very small quantities, such as reservoir volumes, microorganism counts, and trace contaminant concentrations.

Worked Example: Writing a small concentration in scientific notation

A concentration of 0.000042 g/L can be written

\[0.000042\ \text{g/L}=4.2\times10^{-5}\ \text{g/L}.\]

Try It. Write 0.0000068 g/L in scientific notation.
Check Your Work \(6.8\times10^{-6}\) g/L.

For real numbers \(a\), \(b\), \(m\) and \(n\), we have

\[ (ab)^n=a^nb^n,\qquad \qquad \left(\frac ab\right)^n=\frac{a^n}{b^n}.\]

For \(a\ne0\), \(a^0=1\) and \(a^{-n}=1/a^n\).

These rules apply to multiplication and division, not addition. In general, \((a+b)^n\ne a^n+b^n\). Negative exponents do not make a value negative. They indicate reciprocals.

Applied Problem: Scaling storage capacity. Two geometrically similar tanks have linear scale factor 1.5. If the smaller tank holds 32 m³, use the cubic scale factor to estimate the larger tank’s capacity.

Check Your Work Volume scales with the cube of the linear factor. The capacity is \(32(1.5)^3=32(3.375)=108\) m³.

Root. A value that produces a given number when raised to a specified power.

Rational exponents connect powers and roots:

For real numbers \(a \not = 0\), and \(m,n \not = 0\), we have \[a^{m/n}=\sqrt[n]{a^m}.\]

When \(n\) is even, a real principal root requires a nonnegative radicand, and \(\sqrt{x^2}=|x|\). To rationalize \(5/\sqrt3\), multiply by \(\sqrt3/\sqrt3\) to obtain \(5\sqrt3/3\).

Worked Example: Rationalizing a denominator

\[\frac5{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}=\frac{5\sqrt3}{3}.\]

Multiplying by \(\sqrt3/\sqrt3\) changes the form but not the value.

Try It. Rationalize \(4/\sqrt5\).
Check Your Work \(4\sqrt5/5\).

Root properties parallel exponent properties when all expressions are defined.

For \(a,b \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\), we have \[\sqrt[n]{ab}=\sqrt[n]a\sqrt[n]b, \qquad \qquad \sqrt[n]{\frac ab}=\frac{\sqrt[n]a}{\sqrt[n]b}.\]

For odd \(n\), \(\sqrt[n]{a^n}=a\). For even \(n\), \(\sqrt[n]{a^n}=|a|\).

1.5.1 Simplifying roots

Look for perfect-power factors before approximating a root. For example,

\[\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2.\]

An exact radical form retains full precision and often simplifies later algebra. A decimal approximation may be added when the application requires a numerical measurement. When variables appear under an even root, absolute value may be required. For instance, \(\sqrt{49x^2}=7|x|\) for real \(x\).

Worked Example: Finding the side of a square settling basin

A square basin has surface area 180 m². If its side length is \(s\), then \(s^2=180\), so

\[s=\sqrt{180}=6\sqrt5\approx13.4\ \text{m}.\]

Only the positive root is meaningful as a length.

Try It. Find the positive side length of a square basin with area 98 m².
Check Your Work \(s=\sqrt{98}=7\sqrt2\approx9.90\) m.

The conjugate method follows the difference-of-squares identity. When a denominator has the form \(A+B\sqrt C\), multiply by its conjugate \(A-B\sqrt C\). Their product is \(A^2-B^2C\), which contains no square root. Multiplying \(2/(7+\sqrt3)\) by \((7-\sqrt3)/(7-\sqrt3)\) produces denominator \(49-3=46\) and numerator \(14-2\sqrt3\). The simplified result is \((7-\sqrt3)/23\).

Worked Example: Rationalizing with a conjugate

Rationalize the denominator:

\[ \frac{4}{3+\sqrt{5}}. \]

The conjugate of \(3+\sqrt{5}\) is \(3-\sqrt{5}\). Multiply the numerator and denominator by the conjugate:

\[ \frac{4}{3+\sqrt{5}} \cdot \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{4(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}. \]

Use the difference-of-squares pattern in the denominator:

\[ \frac{4(3-\sqrt{5})}{3^2-(\sqrt{5})^2} = \frac{4(3-\sqrt{5})}{9-5} = 3-\sqrt{5}. \]

Therefore,

\[ \frac{4}{3+\sqrt{5}}=3-\sqrt{5}. \]

Try It. Rationalize the denominator and simplify:

\[ \frac{5}{\sqrt{6}+1}. \]

Check Your Work

The conjugate of \(\sqrt{6}+1\) is \(\sqrt{6}-1\). Therefore,

\[ \frac{5}{\sqrt{6}+1} \cdot \frac{\sqrt{6}-1}{\sqrt{6}-1} = \frac{5(\sqrt{6}-1)}{(\sqrt{6})^2-1^2}. \]

Simplify:

\[ \frac{5(\sqrt{6}-1)}{6-1} = \frac{5(\sqrt{6}-1)}{5} = \sqrt{6}-1. \]

Therefore,

\[ \frac{5}{\sqrt{6}+1}=\sqrt{6}-1. \]

1.5.2 Practice Problems

  1. Simplify \(y^4y^3\).
    Check Your Work \(y^7\).
  2. Rewrite \(z^{-3}\) without a negative exponent.
    Check Your Work \(1/z^3\), \(z\ne0\).
  3. Write \(\sqrt[3]{x^2}\) with a rational exponent.
    Check Your Work \(x^{2/3}\).
  4. Simplify \(\sqrt{49a^2}\).
    Check Your Work \(7|a|\).
  5. Rationalize \(3/\sqrt2\).
    Check Your Work \(3\sqrt2/2\).
  6. A bacterial count is \(3.4\times10^5\) cells/mL. Write the count in ordinary notation.
    Check Your Work 340,000 cells/mL.
  7. A square tank cover has area 72 m². Give its exact side length in simplified radical form.
    Check Your Work \(\sqrt{72}=6\sqrt2\) m.
  8. A cylindrical tank’s dimensions are doubled. By what factor does its volume change?
    Check Your Work Volume changes by \(2^3=8\), so it is eight times as large.