Educational

Proof of exponent identity Prove: \[ (x^{a-b})^{a+b}(x^{b-c})^{b+c}(x^{c-a})^{c+a} = 1 \] Proof \[ = x^{(a-b)(a+b)} \cdot x^{(b-c)(b+c)} \cdot x^{(c-a)(c+a)} \] \[ = x^{(a^2-b^2) + (b^2-c^2) + (c^2-a^2)} \] \[ = x^0 \] \[ = 1 \] Hence Proved Next Question / Full Exercise

Simplify (1^3 + 2^3 + 3^3)^(1/2) Simplify: \((1^3 + 2^3 + 3^3)^{1/2}\) Solution \[ = (1 + 8 + 27)^{1/2} \] \[ = (36)^{1/2} \] \[ = 6 \] Final Answer: \[ \boxed{6} \] Next Question / Full Exercise

Simplify 64^(-1/3)(64^(1/3) – 64^(2/3)) Simplify: \(64^{-1/3}(64^{1/3} – 64^{2/3})\) Solution \[ 64 = 4^3 \Rightarrow 64^{1/3} = 4 \] \[ = 64^{-1/3}(4 – 16) \] \[ = \frac{1}{4} \cdot (-12) \] \[ = -3 \] Final Answer: \[ \boxed{-3} \] Next Question / Full Exercise

Simplify exponential expression Simplify: \[ \left[\frac{5^{-1}\cdot7^2}{5^2\cdot7^{-4}}\right]^{7/2} \times \left[\frac{5^{-2}\cdot7^3}{5^3\cdot7^{-5}}\right]^{-5/2} \] Solution \[ \left[\frac{5^{-1}}{5^2}\cdot\frac{7^2}{7^{-4}}\right]^{7/2} = \left[5^{-3}\cdot7^{6}\right]^{7/2} \] \[ = 5^{-21/2}\cdot7^{21} \] \[ \left[\frac{5^{-2}}{5^3}\cdot\frac{7^3}{7^{-5}}\right]^{-5/2} = \left[5^{-5}\cdot7^{8}\right]^{-5/2} \] \[ = 5^{25/2}\cdot7^{-20} \] \[ \text{Multiplying: } = 5^{-21/2+25/2}\cdot7^{21-20} \] \[ = 5^{2}\cdot7 \] \[ = 25\cdot7 = 175 \] Final Answer: \[ \boxed{175} \] Next Question / Full Exercise

Simplify (√2/5)^2 / (√2/5)^13 Simplify: \(\frac{(\sqrt{2}/5)^2}{(\sqrt{2}/5)^{13}}\) Solution \[ = \left(\frac{\sqrt{2}}{5}\right)^{2-13} \] \[ = \left(\frac{\sqrt{2}}{5}\right)^{-11} \] \[ = \left(\frac{5}{\sqrt{2}}\right)^{11} \] Final Answer: \[ \boxed{\left(\frac{5}{\sqrt{2}}\right)^{11}} \] Next Question / Full Exercise

Simplify 5(8^(1/3) + 27^(1/3))^3 Simplify: \(5(8^{1/3} + 27^{1/3})^3\) Solution \[ 8^{1/3} = 2,\quad 27^{1/3} = 3 \] \[ = 5(2 + 3)^3 \] \[ = 5 \cdot 5^3 \] \[ = 5 \cdot 125 \] \[ = 625 \] Final Answer: \[ \boxed{625} \] Next Question / Full Exercise

Simplify (0.001)^(1/3) Simplify: \((0.001)^{1/3}\) Solution \[ 0.001 = 10^{-3} \] \[ = (10^{-3})^{1/3} \] \[ = 10^{-1} \] \[ = 0.1 \] Final Answer: \[ \boxed{0.1} \] Next Question / Full Exercise

Simplify cube root of (343)^(-2) Simplify: \(\sqrt[3]{(343)^{-2}}\) Solution \[ = (343)^{-2/3} \] \[ = (7^3)^{-2/3} \] \[ = 7^{-2} \] \[ = \frac{1}{49} \] Final Answer: \[ \boxed{\frac{1}{49}} \] Next Question / Full Exercise

Simplify 5th root of (32)^(-3) Simplify: \(\sqrt[5]{(32)^{-3}}\) Solution \[ = (32)^{-3/5} \] \[ = (2^5)^{-3/5} \] \[ = 2^{-3} \] \[ = \frac{1}{8} \] Final Answer: \[ \boxed{\frac{1}{8}} \] Next Question / Full Exercise

Proof of a^(q-r) b^(r-p) c^(p-q) = 1 Given \(a=xy^{p-1},\; b=xy^{q-1},\; c=xy^{r-1}\), prove: \[ a^{q-r} b^{r-p} c^{p-q} = 1 \] Proof \[ = (xy^{p-1})^{q-r} (xy^{q-1})^{r-p} (xy^{r-1})^{p-q} \] \[ = x^{q-r}y^{(p-1)(q-r)} \cdot x^{r-p}y^{(q-1)(r-p)} \cdot x^{p-q}y^{(r-1)(p-q)} \] \[ = x^{(q-r)+(r-p)+(p-q)} \cdot y^{(p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q)} \] \[ = x^0 \cdot y^0 \] \[ = 1 \] Hence Proved Next Question /