Mathematics
Operations on real numbers and the laws of exponents
वास्तविक संख्याओं पर संक्रियाएँ और घातांक के नियम
In this Class 10 Mathematics topic from the Polynomials chapter, students strengthen their understanding of operations on real numbers and the laws of exponents. They learn to add, subtract, multiply and divide numerical expressions accurately, use exponent rules for products, quotients and powers, and simplify expressions involving positive, zero and negative exponents where appropriate. The topic builds fluency in working with polynomial terms, comparing equivalent forms, and checking calculations through properties such as commutativity, associativity and distributivity. Examples connect numerical rules with algebraic manipulation and related exercises.
Practice questions
If \(a=2+\sqrt{3}\), what is the value of \(a^{2}+\frac{1}{a^{2}}\)?What is the simplified form of \(\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)+\sqrt{20}\)?If (5^{x}=125) and (2^{y}=32), what is the value of (x+y)?What is the value of (\frac{3^{-2}+3^{-1}}{3^{-3}})?Which option correctly writes \(\left(x^{2}-4\right)\)?If (m=\sqrt{6}+\sqrt{2}), what is the value of (m^{2})?If (r=10^{2}\cdot10^{-5}\cdot10^{4}), what is the value of (r)?What is the simplified form of \(\left(\frac{a^{3}b^{-2}}{a^{-1}b^{2}}\right)^{2}\)?If (u=\frac{\sqrt{3}}{\sqrt{12}}), what is the value of (u^{-2})?If \(a\) is a nonzero real number, which of the following statements is always true?If (x=2-\sqrt{3}), what is the value of (\frac{1}{x}+x)?What is the simplified value of (\frac{6^{5}}{2^{3}\cdot3^{4}})?If \(\left(x^{2}y^{3}\right)^{n}=x^{8}y^{12}\), what is the value of (n)?What is the simplified form of (\sqrt[3]{64x^{6}}), where (x) is real?Which option gives the correct expansion of \(\left(\sqrt{11}-\sqrt{2}\right)^{2}\)?If (2^{a}=8) and (3^{b}=81), what is the value of (a^{b})?What is the simplified form of (\frac{x^{5}-x^{3}}{x^{3}}), where (x\neq0)?What is the value of \(\left(16^{-\frac{3}{4}}\right)\cdot8^{\frac{2}{3}}\)?If (A=7+4\sqrt{3}), what is the correct simplified form of (\sqrt{A})?Which option gives the simplified form of (\frac{x^{-1}+y^{-1}}{(xy)^{-1}}), where (x\neq0) and (y\neq0)?