8.1 Simplify Expressions with Roots

Roots are essential in algebra, allowing us to solve equations and simplify expressions. They represent the inverse of exponents, giving us a way to "undo" powers and find hidden values.

Simplifying expressions with roots involves working with square roots, higher-order roots, and radical expressions. We'll learn to simplify, estimate, and manipulate these expressions, connecting rational and irrational numbers in the process.

Simplifying Expressions with Roots

Simplification of root expressions

• Square roots
  • Definition: $\sqrt{x}$ represents a number that when multiplied by itself equals $x$
  • Simplifying square roots involves
    • Perfect squares: $\sqrt{x^2} = x$ for $x \geq 0$ (4, 9, 16)
    • Factoring out perfect squares: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ if $a$ and $b$ are perfect squares (36, 64)
• Higher-order roots
  • Definition: $\sqrt[n]{x}$ represents a number that when raised to the power of $n$ equals $x$
  • Simplifying higher-order roots involves
    • Perfect $n$th powers: $\sqrt[n]{x^n} = x$ for $x \geq 0$ when $n$ is odd and for all $x$ when $n$ is even (8, 27)
    • Factoring out perfect $n$th powers: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ if $a$ and $b$ are perfect $n$th powers (125, 216)
  • Roots can be expressed using fractional exponents: $\sqrt[n]{x} = x^{\frac{1}{n}}$

Estimation of irrational roots

• Estimating irrational roots involves
  • Identifying the perfect squares or perfect $n$th powers closest to the radicand (2 and 3 for $\sqrt{2.5}$)
  • Estimating the value between the two closest perfect powers ($\sqrt{2.5}$ is between $\sqrt{2}$ and $\sqrt{3}$)
• Calculating approximate values involves
  • Using a calculator to find decimal approximations of irrational roots ($\sqrt{2} \approx 1.4142$)
  • Rounding the decimal approximation to the desired number of decimal places ($\sqrt{2} \approx 1.41$ rounded to 2 decimal places)

Manipulation of radical expressions

• Simplifying radicals with variables involves
  • Identifying perfect square or perfect $n$th power factors within the radicand ($\sqrt{4x^2}$ has a perfect square factor of $4x^2$)
  • Factoring out the perfect power factors and simplifying ($\sqrt{4x^2} = 2|x|$)
• Combining like radicals involves
  • Adding or subtracting radicals with the same index and radicand ($\sqrt{2} + \sqrt{2} = 2\sqrt{2}$)
  • Simplifying the result if possible ($2\sqrt{4} = 4$)
• Multiplying radicals involves
  • Multiplying the radicands under the same root index ($\sqrt{2} \cdot \sqrt{3} = \sqrt{6}$)
  • Simplifying the result if possible ($\sqrt{9} = 3$)
• Dividing radicals involves
  • Rewriting division as multiplication by the reciprocal of the divisor ($\frac{\sqrt{2}}{\sqrt{3}} = \sqrt{2} \cdot \frac{1}{\sqrt{3}}$)
  • Multiplying the radicands under the same root index ($\sqrt{2} \cdot \frac{1}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{3}}$)
  • Simplifying the result if possible ($\frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3}$)
• Rationalizing denominators involves
  • Multiplying the numerator and denominator by the conjugate of the denominator ($\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$)
  • Simplifying the result to eliminate the radical in the denominator ($\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$)

Number Systems and Roots

• Rational numbers: numbers that can be expressed as a ratio of two integers (e.g., $\frac{1}{2}$, $-3$, $0.75$)
• Irrational numbers: numbers that cannot be expressed as a ratio of two integers (e.g., $\sqrt{2}$, $\pi$)
• Real numbers: the set of all rational and irrational numbers
• Radicals often produce irrational numbers, but not always (e.g., $\sqrt{4} = 2$ is rational)