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📘intermediate algebra review

key term - Perfect nth Powers

Citation:

Definition

A perfect nth power is a number that can be expressed as the nth power of an integer. In other words, it is a number that is the result of raising a positive integer to a positive integer power.

5 Must Know Facts For Your Next Test

  1. Perfect nth powers can be simplified by rewriting them as the nth root of the number.
  2. The square root of a perfect square is the number itself, raised to the power of 1/2.
  3. The cube root of a perfect cube is the number itself, raised to the power of 1/3.
  4. Perfect nth powers are often used in simplifying expressions with roots, as they can be reduced to a single term.
  5. Recognizing perfect nth powers is important for efficiently simplifying algebraic expressions involving roots.

Review Questions

  • Explain how perfect nth powers can be used to simplify expressions with roots.
    • When an expression contains a perfect nth power, it can be simplified by rewriting the expression using the nth root. For example, $x^4$ is a perfect fourth power, so it can be rewritten as $\sqrt[4]{x^4} = x$. This allows the expression to be reduced to a single term, making it easier to evaluate and manipulate.
  • Describe the relationship between perfect squares and their square roots.
    • A perfect square is a number that can be expressed as the product of an integer multiplied by itself. The square root of a perfect square is the number itself raised to the power of 1/2. For instance, the square root of 16 is 4, because $\sqrt{16} = 4^{1/2} = 4$. This property of perfect squares is useful for simplifying expressions involving square roots.
  • Analyze how recognizing perfect nth powers can improve the efficiency of simplifying algebraic expressions.
    • Identifying perfect nth powers within an algebraic expression allows you to rewrite the expression in a more simplified form. This can significantly reduce the number of steps required to simplify the expression and make it easier to work with. By recognizing perfect nth powers, you can avoid unnecessary calculations and manipulations, leading to a more efficient simplification process. This skill is particularly valuable when dealing with complex expressions involving multiple roots or powers.