Question

Rewrite each expression in simplest form.
(a) $6\times x\times y$ (b) $7\times a\times b$ (c) $x\times y\times z$ (d) $2\times y\times y$
(e) $a\times 4\times b$ (f) $x\times y\times 12$ (g) $5\times b\times a$ (h) $y\times z\times z$
(i) $6+x$ (j) $4x\div 2y$ (k) $\left( x+3 \right)\div 4$ (l) $m\times m\times m\div m\times m$
(m) $4\times x+5\times y$ (n) $a\times 7-2\times b$ (o) $2\times x\times \left( x-4 \right)$ (p) $3\times \left( x+1 \right)\div 2\times x$
(q) $2\times \left( x+4 \right)\div 3$ (r) $\left( 4\times x \right)\div \left( 2\times x+4\times x \right)$

Answer 1

Hint: Simplify each of the given algebraic expressions by following the rule of BODMAS. Use the law of exponents which states that ${{a}^{b}}\times {{a}^{c}}={{a}^{b+c}}$ and $\dfrac{{{a}^{b}}}{{{a}^{c}}}={{a}^{b-c}}$. Keep in mind that the coefficient of two terms can be added or subtracted if both the terms have the same variable.

Complete step-by-step solution -
We have to simplify each of the given algebraic expressions.
To do so, we will follow the rules of BODMAS and law of exponents which state that ${{a}^{b}}\times {{a}^{c}}={{a}^{b+c}}$ and $\dfrac{{{a}^{b}}}{{{a}^{c}}}={{a}^{b-c}}$.
We can add or subtract the coefficients of two terms only if both the terms have the same variables.
We will now simplify each of the given options.
(a) Multiplying all the terms of the expression $6\times x\times y$, we can rewrite it as $6xy$.
(b) Multiplying all the terms of the expression $7\times a\times b$, we can rewrite it as $7ab$.
(c) Multiplying all the terms of the expression $x\times y\times z$, we can rewrite it as $xyz$.
(d) Multiplying all the terms of the expression $2\times y\times y$, we can rewrite it as $2{{y}^{2}}$.
(e) Multiplying all the terms of the expression $a\times 4\times b$, we can rewrite it as $4ab$.
(f) Multiplying all the terms of the expression $x\times y\times 12$, we can rewrite it as $12xy$.
(g) Multiplying all the terms of the expression $5\times b\times a$, we can rewrite it as $5ab$.
(h) Multiplying all the terms of the expression $y\times z\times z$, we can rewrite it as $y{{z}^{2}}$.
(i) We can’t further simplify the expression $6+x$. So, we can write it as $6+x$.
(j) Simplifying the expression $4x\div 2y$, we can rewrite it as $\dfrac{4x}{2y}=\dfrac{2x}{y}$.
(k) Simplifying the expression $\left( x+3 \right)\div 4$, we can rewrite it as $\dfrac{x+3}{4}$.
(l) Simplifying the expression $m\times m\times m\div m\times m$ using BODMAS, we can rewrite it as $m\times m\times \left( \dfrac{m}{m} \right)\times m=m\times m\times 1\times m={{m}^{3}}$.
(m) Simplifying the expression $4\times x+5\times y$, we can rewrite it as $4x+5y$.
(n) Simplifying the expression $a\times 7-2\times b$, we can rewrite it as $7a-2b$.
(o) Multiplying all the terms of the expression $2\times x\times \left( x-4 \right)$, we can rewrite it as $2x\times \left( x-4 \right)=2{{x}^{2}}-8x$.
(p) Simplifying the expression $3\times \left( x+1 \right)\div 2\times x$ using BODMAS, we can rewrite it as $3\times \left( \dfrac{x+1}{2} \right)\times x=\dfrac{3x\left( x+1 \right)}{2}=\dfrac{3{{x}^{2}}+3x}{2}$.
(q) Simplifying the expression $2\times \left( x+4 \right)\div 3$, we can rewrite it as $2\times \left( \dfrac{x+4}{3} \right)=\dfrac{2x+8}{3}$.
(r) Simplifying the expression $\left( 4\times x \right)\div \left( 2\times x+4\times x \right)$ using BODMAS, we can rewrite it as $\left( 4x \right)\div \left( 2x+4x \right)=\left( 4x \right)\div \left( 6x \right)=\dfrac{4x}{6x}=\dfrac{2}{3}$.

Note: We can’t solve this question without using the law of exponents. If we will now apply BODMAS and randomly apply the operations, we will get an incorrect answer. BODMAS says that we must simplify the operations in the order- Bracket, Of, Division, Multiplication, Addition, and Subtraction.