Podcast
Questions and Answers
Rationalize the denominator of $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$.
Rationalize the denominator of $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$.
$5 - 2\sqrt{6}$
Simplify: $\left(\frac{3}{4}\right)^{-2} + \left(\frac{4}{9}\right) \times \frac{16}{27}$
Simplify: $\left(\frac{3}{4}\right)^{-2} + \left(\frac{4}{9}\right) \times \frac{16}{27}$
$\frac{253}{81}$
Simplify: $\sqrt[7]{\frac{x^{14} \cdot y^{21} \cdot z^{35}}{y^{14} \cdot z^{7}}}$
Simplify: $\sqrt[7]{\frac{x^{14} \cdot y^{21} \cdot z^{35}}{y^{14} \cdot z^{7}}}$
$\frac{x^2y}{z^4}$
Simplify: $\frac{5 \cdot (25)^{n+1} - 25 \cdot (5)^{2n}}{5 \cdot (5)^{2n+3} - (25)^{n+1}}$
Simplify: $\frac{5 \cdot (25)^{n+1} - 25 \cdot (5)^{2n}}{5 \cdot (5)^{2n+3} - (25)^{n+1}}$
Simplify the expression: $\frac{3^n \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$
Simplify the expression: $\frac{3^n \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}$
If $x = 3 + \sqrt{8}$, find the value of $\frac{1}{x}$.
If $x = 3 + \sqrt{8}$, find the value of $\frac{1}{x}$.
Determine the values of rational numbers $p$ and $q$ such that $\frac{8 - 3\sqrt{2}}{4 + 3\sqrt{2}} = p + q\sqrt{2}$.
Determine the values of rational numbers $p$ and $q$ such that $\frac{8 - 3\sqrt{2}}{4 + 3\sqrt{2}} = p + q\sqrt{2}$.
Simplify the following expression: $\frac{54 \times \sqrt[3]{(27)^{2x}}}{9^{x+1} + 216(3^{2x-1})}$
Simplify the following expression: $\frac{54 \times \sqrt[3]{(27)^{2x}}}{9^{x+1} + 216(3^{2x-1})}$
Flashcards
Rationalize the Denominator
Rationalize the Denominator
To express a fraction with a rational denominator.
Rational Number
Rational Number
A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Simplifying Exponents
Simplifying Exponents
Simplifying expressions with exponents involves applying exponent rules to reduce them to their simplest form.
Irrational Number
Irrational Number
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Solving Equations
Solving Equations
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Simplifying
Simplifying
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Study Notes
- These notes cover mathematical expressions and simplifications.
Simplifying Expressions with Radicals
- (iv) simplifies (6-4√2) / (6+4√2).
- (v) simplifies (√3-√2) / (√3+√2).
- (vi) simplifies 4√3 / (√7+√5).
Simplifying Expressions with Exponents
- (i) simplifies (81/16)^(-3/4).
- (ii) simplifies (3/4)^(-2) + (4/9) * (16/27).
- (iii) simplifies (0.027)^(-1/3).
- (iv) simplifies √(x^14 * y^21 * z^35) / (y^14 * z^7).
- (v) simplifies [5 * (25)^(n+1) - 25 * (5)^(2n)] / [5 * (5)^(2n+3) - (25)^(n+1)].
- (vi) simplifies (16^(x+1) + 20 * (4^(2x))) / (2^(x-3) * 8^(x+2)).
- (vii) simplifies (64)^(2/3) ÷(9)^(-3/2).
- (viii) simplifies (3^n * 9^(n+1)) / (3^(n-1) * 9^(n-1)).
- (ix) simplifies (5^(n+3) - 6 * 5^(n+1)) / (9 * 5^n - 2^n * 5^n).
Finding Values Based on a Given Condition
- Given x = 3 + √8, find the value of:
- (i) x + 1/x
- (ii) x - 1/x
- (iii) x² + 1/x²
- (iv) x² - 1/x²
- (v) x⁴ + 1/x⁴
- (vi) (x - 1/x)²
Rational Numbers
- Find rational numbers p and q such that (8 - 3√2) / (4 + 3√2) = p + q√2.
Simplifying Expressions
- (i) simplifies [(25)^3 * (243)^5] / [(16)^4 * (8)^3].
- (ii) simplifies [54 * ∛(27)^(2x)] / [9^(x+1) + 216 * (3^(2x-1))].
- (iii) simplifies √[(216)^(2/3) * (25)^(1/2)] / (0.04)^(-3/2).
- (iv) simplifies (a^(1/3) + b^(2/3)) * (a^(2/3) - a^(1/3)b^(2/3) + b^(4/3)).
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