Indices

Find the value of $3^5 \times 3^{-5}$.

Find the value of $9^4 \times \dfrac{1}{9^3}$.

Find the value of $5^3 \times \dfrac{1}{5^2}$.

Find the value of $64^{2/3}$.

Find the value of $\left(\dfrac{81}{3}\right)^{1/3}$.

Find the value of $\sqrt[3]{\dfrac{1}{125}}$.

Find the value of $\left(\dfrac{27}{64}\right)^{-2/3}$.

Find the value of $\left\{\left(\dfrac{216}{125}\right)^{-1/3}\right\}^2$.

Find the value of $\left(\dfrac{1}{128}\right)^{1/7} + \left(\dfrac{1}{64}\right)^{1/6}$.

Find the value of $\left(\dfrac{1}{16}\right)^{1/4} \times 8^{4/3}$.

Find the value of $\left(\dfrac13\right)^{-3} + \left(\dfrac13\right)^{-1} + \left(\dfrac12\right)^{-3} + \left(\dfrac18\right)^0$.

Simplify $\left(125m^3 \div 27b^{-3}\right)^{-2/3}$.

Simplify $\left(64a^3 \div 125b^{-3}\right)^{-2/3}$.

Simplify $x^{a - b} \times x^{b - c} \times x^{c - a}$.

Simplify $x^{b^2 - c^2} \times x^{c^2 - a^2} \times x^{a^2 - b^2}$.

Simplify $x^{ab} \times x^{bc} \times x^{ca} \times x^{-ab} \times x^{-bc} \times x^{-ca}$.

Simplify $\dfrac{x^{2m + 3n} \times x^{3m + 6n}}{x^{2m + 3n} \times x^{4m - 4n}}$.

Simplify $\dfrac{a^{9n + 3} \times a^{-4n}}{a^{2n + 10} \times a^{3n - 7}}$.

Simplify $\dfrac{7^{n + 1} \times 7^{n^2 - 1}}{7^{n^2 - n} \times 7^{2n + 2}}$.

Find the value of $\left(\dfrac{27}{8}\right)^{-1/3}\left[\left(\dfrac{81}{16}\right)^{1/4} \div \left(\dfrac{4}{25}\right)^{-1/2}\right]$.

Find the value of $\left(\dfrac{25}{16}\right)^{-1/2}\left[\left(\dfrac{125}{64}\right)^{1/3} \div \left(\dfrac{8}{27}\right)^{-1/3}\right]$.

Find the value of $\left(\dfrac{27}{8}\right)^{1/3}\left[\left(\dfrac{243}{32}\right)^{1/5} \div \left(\dfrac23\right)^{-2}\right]$.

Find the value of $\dfrac{3^4 \times 27^3 \times 9^5}{81^6 \times 3^3 \times 9^{-2}}$.

Find the value of $\dfrac{2^4 \times 8^3 \times 4^5}{16^6 \times 2^3 \times 4^{-2}}$.

Simplify $\dfrac{1}{1 + x^{a - b}} + \dfrac{1}{1 + x^{b - a}}$.

Simplify $\dfrac{1}{1 - a^{x - y}} + \dfrac{1}{1 - a^{y - x}}$.

Simplify $\dfrac{(x^{a + b})^2 \times (x^{b + c})^2 \times (x^{c + a})^2}{(x^a \cdot x^b \cdot x^c)^4}$.

Simplify $\dfrac{(a^{x + y})^3 \times (a^{y + z})^3 \times (a^{z + x})^3}{(a^x \cdot a^y \cdot a^z)^6}$.

Simplify $\dfrac{27^{3a + 1} \times 243^{-4a/5}}{9^{a + 5} \times 3^{3a - 7}}$.

Simplify $\dfrac{32^{2n/5} \times 2^{2n + 1}}{4^{n + 1} \times 2^{2(n - 2)}}$.

Simplify $\left(\dfrac{x^b}{x^{-a}}\right)^{b - a} \times \left(\dfrac{x^a}{x^{-c}}\right)^{a - c} \times \left(\dfrac{x^c}{x^{-b}}\right)^{c - b}$.

Simplify $\left(\dfrac{x^{-b}}{x^{-a}}\right)^{a + b} \times \left(\dfrac{x^{-c}}{x^{-b}}\right)^{c + b} \times \left(\dfrac{x^{-a}}{x^{-c}}\right)^{c + a}$.

Simplify $\left(\dfrac{x^m}{x^{-n}}\right)^{m^2 - mn + n^2} \times \left(\dfrac{x^n}{x^{-l}}\right)^{n^2 - nl + l^2} \times \left(\dfrac{x^l}{x^{-m}}\right)^{l^2 - ml + m^2}$.

Simplify $\left(\dfrac{x^{a + b}}{x^c}\right)^{a - b} \times \left(\dfrac{x^{b + c}}{x^a}\right)^{b - c} \times \left(\dfrac{x^{c + a}}{x^b}\right)^{c - a}$.

Simplify $\sqrt[a + b]{x^{a^2 - b^2}} \times \sqrt[b + c]{x^{b^2 - c^2}} \times \sqrt[c + a]{x^{c^2 - a^2}}$.

Simplify $\left(\dfrac{x^{a^2 + b^2}}{x^{-ab}}\right)^{a - b} \times \left(\dfrac{x^{b^2 + c^2}}{x^{-bc}}\right)^{b - c} \times \left(\dfrac{x^{c^2 + a^2}}{x^{-ca}}\right)^{c - a}$.

Simplify $\left(\dfrac{x^b}{x^c}\right)^{1/(bc)} \times \left(\dfrac{x^c}{x^a}\right)^{1/(ca)} \times \left(\dfrac{x^a}{x^b}\right)^{1/(ab)}$.

Simplify $\left\{(a^x \cdot a^y)^{x - y}\left(\dfrac{a^y}{a^z}\right)^{y + z}\right\} \times \left(\dfrac{a^z}{a^x}\right)^{z + x}$.

Prove that $\sqrt[bc]{\dfrac{x^b}{x^c}} \times \left(\dfrac{x^c}{x^a}\right)^{1/(ca)} \times \sqrt[ab]{\dfrac{x^a}{x^b}} = 1$.

Prove that $\sqrt{\dfrac{x^a}{x^b}} \times \sqrt{\dfrac{x^b}{x^c}} \times \sqrt{\dfrac{x^c}{x^a}} = 1$.
The textbook prints the roots as the c-th, a-th and b-th. Then the index is $\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b}$, which is not 0 in general (a = 1, b = 2, c = 3 gives $-\frac13$), so the book’s numbers don’t work. With square roots for all three, as the Nepali edition prints it and as here, the statement is true.

Prove that $\sqrt[bc]{\dfrac{x^{b/c}}{x^{c/b}}} \times \sqrt[ca]{\dfrac{x^{c/a}}{x^{a/c}}} \times \sqrt[ab]{\dfrac{x^{a/b}}{x^{b/a}}} = 1$.

Prove that $\dfrac{1}{1 + x^{a - b} + x^{c - b}} + \dfrac{1}{1 + x^{b - c} + x^{a - c}} + \dfrac{1}{1 + x^{c - a} + x^{b - a}} = 1$.

Prove that $\dfrac{\left(p + \frac1q\right)^m \times \left(p - \frac1q\right)^m}{\left(q + \frac1p\right)^m \times \left(q - \frac1p\right)^m} = \left(\dfrac pq\right)^{2m}$.

Prove that $\dfrac{\left(x^2 - \frac{1}{y^2}\right)^a \times \left(x - \frac1y\right)^{b - a}}{\left(y^2 - \frac{1}{x^2}\right)^b \times \left(y + \frac1x\right)^{a - b}} = \left(\dfrac xy\right)^{a + b}$.
The textbook prints $x^2 + \frac{1}{y^2}$ and $y^2 + \frac{1}{x^2}$. These don’t factorize, and then the two sides are not equal (x = 3, y = 2, a = 1, b = 2 gives about 3.19 against 3.375), so the book’s numbers don’t work. With minus signs, as the Nepali edition prints them and as in Example 5, they are.

Prove that $\dfrac{\left(a^2 - \frac{1}{b^2}\right)^x \times \left(a - \frac1b\right)^{y - x}}{\left(b^2 - \frac{1}{a^2}\right)^y \times \left(b + \frac1a\right)^{x - y}} = \left(\dfrac ab\right)^{x + y}$.

If $p^3 + q^3 + r^3 = 1$, prove that $\left(\dfrac{x^p}{x^{-q}}\right)^{p^2 - pq + q^2} \times \left(\dfrac{x^q}{x^{-r}}\right)^{q^2 - qr + r^2} \times \left(\dfrac{x^r}{x^{-p}}\right)^{r^2 - rp + p^2} = x^2$.

If $x^2 + y^2 + z^2 = xy + yz + zx$, prove that $\left(\dfrac{a^x}{a^y}\right)^{x - y} \times \left(\dfrac{a^y}{a^z}\right)^{y - z} \times \left(\dfrac{a^z}{a^x}\right)^{z - x} = 1$.
The textbook prints the condition as $x^2 + y^2 + z^2 = 2(xy + yz + zx)$. Then the left side comes to $a^{2(xy + yz + zx)}$, not 1, so the book’s numbers don’t work. Without the 2, as the Nepali edition prints it and as here, it is 1.

If g + h + f = 0, prove that $\dfrac{1}{1 + m^g + m^{-h}} + \dfrac{1}{1 + m^h + m^{-f}} + \dfrac{1}{1 + m^f + m^{-g}} = 1$.

If r + s + t = 0, prove that $\dfrac{1}{1 + x^r + x^{-s}} + \dfrac{1}{1 + x^s + x^{-t}} + \dfrac{1}{1 + x^t + x^{-r}} = 1$.

If xyz = 1, prove that $\dfrac{1}{1 + x + y^{-1}} + \dfrac{1}{1 + y + z^{-1}} + \dfrac{1}{1 + z + x^{-1}} = 1$.

Factorize $216a^3 + \dfrac18$.

Factorize $8x^3 - 125y^3$.

Factorize $(x + 2)^3$.

Factorize $(x - 3)^3$.

Factorize $16m^4 - 65m^2n^2 + 49n^4$.

Factorize $9x^4 - \dfrac{2x^2}{y^2} + \dfrac{1}{9y^4}$.

Factorize $64 - 144a + 108a^2 - 27a^3$.

Factorize $a^4 - 5a^2b^2 + 4b^4$.

Find the HCF of $x^2 + 2x - 8 - y^2 - 6y$, $x^2 + 2xy + y^2 - 16$ and $x^2 + xy + 4x$.

Find the HCF of $9(a + b)^2 + a + b - 8$, $a^2 + 2ab + b^2 - 1$ and $a^2 + ab + a$.

Find the HCF of $x^2 - 10x + 24 + 6y - 9y^2$, $x^2 + 6xy + 9y^2 - 36$ and $x^2 + 3xy - 6x$.

Find the HCF of $\dfrac{x^4}{y^4} + 1 + \dfrac{y^4}{x^4}$, $\dfrac{x^3}{y^3} + \dfrac{y^3}{x^3}$ and $\dfrac{x^3}{y^2} - x + \dfrac{y^2}{x}$.

Find the LCM of $a^5 + 2a^4 - 9a^3 - 18a^2$, $a^5 - 2a^4 + 8a^2 - 4a^3$ and $a^4 - 4a^2$.

Find the LCM of $x^4 - 8x^2 - 33 - 14y - y^2$, $(x^2 + y)^2 - 9$ and $x^3 + xy + 3x$.

Find the LCM of $x^2 + 2xy + y^2 - z^2$, $y^2 + 2yz + z^2 - x^2$ and $z^2 + 2zx + x^2 - y^2$.

Find the LCM of a(a + c) − b(b + c), b(a + b) − c(c + a) and c(b + c) − a(a + b).

The HCF and LCM of two algebraic expressions are 2u + 5v and 3u(2u + 5v)(2u − 5v). If one expression is $4u^2 - 25v^2$, find the other.

Find the HCF and LCM of $(p + 3)^3$, $p^2 + 6p + 9$ and $2p^3 + 18p^2 + 54p + 54$.

Find the HCF and LCM of $q^2 - 6q - 40 + 14r - r^2$, $q^2 - 2qr + r^2 - 16$ and $q^2 - qr + 4q$.
The textbook prints the third expression as $q^2 - qr + 4r$, which has no factor in common with the other two, so the book’s numbers don’t fit its own answer. $q^2 - qr + 4q = q(q - r + 4)$, as here, does.

Solve by substitution: 2x + 3y = 8, 3x + 2y = 7.

Solve by substitution: $\dfrac23x + 2y = 1$, $\dfrac x3 - \dfrac y3 = 1$.

Solve by substitution: 5x − 2y = 10, 4x − 3y = −6.

Solve by elimination: $\dfrac x6 - y = -6$, $\dfrac{3x}{4} - 1 = y$.

Solve by elimination: $\dfrac{x + y}{2} = \dfrac{3x - 5y}{4}$, $\dfrac{x - 3y}{2} = \dfrac{x - 1}{3}$.

Muna’s present age is 4 years more than twice Umesh’s. Umesh’s present age is 2 years more than $\dfrac25$ of Muna’s. Find the difference between their ages.

The present ages of Tilak and Bimala are in the ratio 5 : 7. Eight years ago the ratio was 7 : 13. Find their present ages.

In a two-digit number the ones digit is three times the tens digit. If 36 is added to the number, its digits change places. Find the number.

Zeenat is setting up a new office. At a shop, 3 tables and 2 chairs cost Rs 1,900, and 2 tables and 4 chairs cost Rs 1,800. Find the price of a table and of a chair, and the cost of one table and four chairs.

Simplify $\left(\dfrac{x^m}{x^n}\right)^{m - p + n} \times \left(\dfrac{x^n}{x^p}\right)^{n - m + p} \times \left(\dfrac{x^p}{x^m}\right)^{p - n + m}$.

Simplify $\dfrac{1}{1 + x^{p - q} + x^{p - r}} + \dfrac{1}{1 + x^{q - p} + x^{q - r}} + \dfrac{1}{1 + x^{r - p} + x^{r - q}}$.

Simplify $\dfrac{\left(9x^2 - \frac{1}{9y^2}\right)^{3x} \left(3x - \frac{1}{3y}\right)^{3y - 3x}}{\left(9y^2 - \frac{1}{9x^2}\right)^{3y} \left(3y + \frac{1}{3x}\right)^{3x - 3y}}$.
The textbook prints the last bracket as $3y + \frac{1}{3y}$. Then nothing cancels and the answer isn’t a single power (x = 2, y = 1 gives about 439 instead of 512), so the book’s numbers don’t work. With $3y + \frac{1}{3x}$, as here, it is Example 5 again.

If $\dfrac{1}{ab} + \dfrac{1}{bc} + \dfrac{1}{ca} = 0$, prove that $(1 + x^a + x^{-b})^{-1} + (1 + x^b + x^{-c})^{-1} + (1 + x^c + x^{-a})^{-1} = 1$.