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Questions and Answers
What is the correct radical form of the expression $x^{2/3}$?
What is the correct radical form of the expression $x^{2/3}$?
- Cube root of $x^2$ (correct)
- Square root of $x^2$
- Fifth root of $x^2$
- Square root of the cube of $x$
Which of the following represents the expression $y^{-2/5}$ in radical form?
Which of the following represents the expression $y^{-2/5}$ in radical form?
- Fifth root of the reciprocal of $y^2$
- Fifth root of $y^{-2}$
- Fifth root of $y^2$
- 1 divided by the fifth root of $y^2$ (correct)
How would you express $x^{3/4}$ using radicals?
How would you express $x^{3/4}$ using radicals?
- Fourth root of the cube of $x$
- Fourth root of $x^3$ (correct)
- Square root of $x^{3/4}$
- Cube root of $x^4$
What is the radical equivalent of $x^{1/2}$?
What is the radical equivalent of $x^{1/2}$?
What does the expression $x^{4/5} * x^{3/5}$ simplify to in radical form?
What does the expression $x^{4/5} * x^{3/5}$ simplify to in radical form?
Flashcards
Rational Exponent
Rational Exponent
A way of expressing mathematical expressions where the base is raised to a fractional power (m/n); xm/n means the base x is raised to the power m and then taken to the n-th root.
Radical
Radical
A way of expressing mathematical expressions where the base is raised to a power and then taken to a specific root; √n(xm) means the base x is raised to the power m and then taken to the n-th root.
Converting between rational exponents and radicals
Converting between rational exponents and radicals
The numerator of a rational exponent corresponds to the exponent of the base inside the radical, and the denominator of the rational exponent corresponds to the index of the radical. For example, x2/3 is equivalent to √3(x2)
Simplifying expressions with rational exponents
Simplifying expressions with rational exponents
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Rational exponents with a 1 in the numerator
Rational exponents with a 1 in the numerator
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Study Notes
Converting Between Rational Exponents and Radicals
- Rational exponents and radicals are different ways to express the same mathematical concept; they are interchangeable.
- A rational exponent, such as xm/n, represents a base (x) raised to a fractional power, where m is the exponent and n is the root.
- A radical, such as √n(xm), represents a base (x) raised to a power (m) and then taken to an n-th root.
Key Relationships
- The numerator of the rational exponent corresponds to the exponent of the base within the radical.
- The denominator of the rational exponent corresponds to the index of the radical.
- Specifically, xm/n is equivalent to √n(xm).
Examples of Conversion
- x1/2 is equivalent to √2(x) or simply √(x) (square root of x).
- x2/3 is equivalent to √3(x2) (cube root of x squared).
- x3/4 is equivalent to √4(x3) (fourth root of x cubed).
- y-2/5 is equivalent to 1/√5(y2) (the fifth root of y squared in the denominator).
Simplifying Expressions with Rational Exponents
- Expressions can be simplified by converting to radical form, carrying out the operations, then converting back to exponential form.
- When simplifying expressions involving rational exponents with a 1 in the numerator, the expression is equal to the n-th root of x.
- Manipulating terms is often necessary before simplification.
Examples of Simplification
- x6/3 can first be simplified as x2.
- Consider simplifying x4/5 * x3/5*. Combining the exponents gives x7/5. This is equivalent to √5(x7).
- If you have an expression like √3(8x6), this can be broken down using properties of exponents. √3(8x6) = √3(23 * (x2)3) = 2x2.
Important Considerations:
- The base (x) must be a non-negative real number if the root is an even number (e.g., √2) or zero if the root is an odd number.
- Negative rational exponents indicate a reciprocal relationship.
- Always attempt to simplify the expression as much as possible before carrying out calculations to reduce errors.
Mixed Form Expressions
- Expressions sometimes combine radicals and rational exponents.
- Approach these problems by first rewriting all terms using either rational exponents or radicals. Carry out the calculations, and then convert back to the desired format.
Common Mistakes
- Incorrectly identifying the numerator and denominator of the rational exponent with the exponent and root in the radical.
- Forgetting to apply the indices (roots) to both powers indicated in the radical notation.
Expressing answers in simplest form
- Ensure the answer is completely simplified with no repeated bases or fractions.
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