Ravi Kant Kumar

Find x Question \[ 2^4 \times 4^2 = 16^x \] Solution \[ 4 = 2^2,\quad 16 = 2^4 \] \[ 2^4 \times (2^2)^2 = (2^4)^x \] \[ 2^4 \times 2^4 = 2^{4x} \] \[ 2^8 = 2^{4x} \] \[ 8 = 4x \] \[ x = 2 \] Answer \[ \boxed{2} \] Next Question / […]

Power Law of Exponents Statement \[ (a^m)^n = a^{mn} \] Condition \[ a \neq 0,\quad m,n \in \mathbb{R} \] Answer \[ \boxed{(a^m)^n = a^{mn}} \] Next Question / Full Exercise

Quotient Law of Exponents Statement \[ \frac{a^m}{a^n} = a^{m-n} \] Condition \[ a \neq 0,\quad m,n \in \mathbb{R} \] Answer \[ \boxed{\frac{a^m}{a^n} = a^{m-n}} \] Next Question / Full Exercise

Product Law of Exponents Statement \[ a^m \times a^n = a^{m+n} \] Condition \[ a \neq 0,\quad m,n \in \mathbb{R} \] Answer \[ \boxed{a^m \cdot a^n = a^{m+n}} \] Next Question / Full Exercise

625^-1/4 in Decimal Form Question \[ 625^{-1/4} \] Solution \[ 625 = 5^4 \] \[ (5^4)^{-1/4} = 5^{-1} \] \[ = \frac{1}{5} \] \[ = 0.2 \] Answer \[ \boxed{0.2} \] Next Question / Full Exercise

Proof of Given Expression Question \[ x=a^{m+n},\quad y=a^{n+l},\quad z=a^{l+m} \] \[ \text{Prove } x^m y^n z^l = x^n y^l z^m \] Solution \[ x^m y^n z^l = a^{m(m+n)} \cdot a^{n(n+l)} \cdot a^{l(l+m)} \] \[ = a^{m^2+mn+n^2+nl+l^2+lm} \] \[ x^n y^l z^m = a^{n(m+n)} \cdot a^{l(n+l)} \cdot a^{m(l+m)} \] \[ = a^{mn+n^2+nl+l^2+lm+m^2} \] \[ \Rightarrow x^m

Proof of Given Expression = 1 Question \[ a=x^{m+n}y^{l},\quad b=x^{n+l}y^{m},\quad c=x^{l+m}y^{n} \] \[ \text{Prove } a^{m-n}b^{n-l}c^{l-m}=1 \] Solution \[ a^{m-n} = x^{(m+n)(m-n)} y^{l(m-n)} \] \[ b^{n-l} = x^{(n+l)(n-l)} y^{m(n-l)} \] \[ c^{l-m} = x^{(l+m)(l-m)} y^{n(l-m)} \] \[ \Rightarrow a^{m-n}b^{n-l}c^{l-m} = x^{(m+n)(m-n)+(n+l)(n-l)+(l+m)(l-m)} \cdot y^{l(m-n)+m(n-l)+n(l-m)} \] \[ = x^{(m^2-n^2)+(n^2-l^2)+(l^2-m^2)} \cdot y^{(lm-ln+mn-ml+nl-nm)} \] \[ = x^0 \cdot y^0

Proof of Given Expression Question \[ \frac{(a+\frac{1}{b})^m (a-\frac{1}{b})^n}{(b+\frac{1}{a})^m (b-\frac{1}{a})^n} \] Solution \[ a+\frac{1}{b} = \frac{ab+1}{b},\quad a-\frac{1}{b} = \frac{ab-1}{b} \] \[ b+\frac{1}{a} = \frac{ab+1}{a},\quad b-\frac{1}{a} = \frac{ab-1}{a} \] \[ = \frac{\left(\frac{ab+1}{b}\right)^m \left(\frac{ab-1}{b}\right)^n}{\left(\frac{ab+1}{a}\right)^m \left(\frac{ab-1}{a}\right)^n} \] \[ = \frac{(ab+1)^m (ab-1)^n}{b^{m+n}} \cdot \frac{a^{m+n}}{(ab+1)^m (ab-1)^n} \] \[ = \frac{a^{m+n}}{b^{m+n}} \] \[ = \left(\frac{a}{b}\right)^{m+n} \] Answer \[ \boxed{\left(\frac{a}{b}\right)^{m+n}} \] Next Question

Simplification of Given Expression Question \[ \sqrt[lm]{\frac{x^l}{x^m}} \cdot \sqrt[mn]{\frac{x^m}{x^n}} \cdot \sqrt[nl]{\frac{x^n}{x^l}} \] Solution \[ = \left(x^{l-m}\right)^{\frac{1}{lm}} \cdot \left(x^{m-n}\right)^{\frac{1}{mn}} \cdot \left(x^{n-l}\right)^{\frac{1}{nl}} \] \[ = x^{\frac{l-m}{lm}} \cdot x^{\frac{m-n}{mn}} \cdot x^{\frac{n-l}{nl}} \] \[ = x^{\left(\frac{l-m}{lm} + \frac{m-n}{mn} + \frac{n-l}{nl}\right)} \] \[ = x^{\left(\frac{1}{m}-\frac{1}{l} + \frac{1}{n}-\frac{1}{m} + \frac{1}{l}-\frac{1}{n}\right)} \] \[ = x^0 \] \[ = 1 \] Answer \[

Simplification of Given Expression Question \[ (x^{a+b}/x^c)^{a-b}(x^{b+c}/x^a)^{b-c}(x^{c+a}/x^b)^{c-a} \] Solution \[ = (x^{a+b-c})^{a-b}(x^{b+c-a})^{b-c}(x^{c+a-b})^{c-a} \] \[ = x^{(a+b-c)(a-b)} \cdot x^{(b+c-a)(b-c)} \cdot x^{(c+a-b)(c-a)} \] \[ = x^{[(a+b-c)(a-b) + (b+c-a)(b-c) + (c+a-b)(c-a)]} \] \[ = x^{(a^2-b^2 – ac + bc + b^2-c^2 – ab + ca + c^2-a^2 – bc + ab)} \] \[ = x^0 \] \[ =