What is the simplified form of ((2m-n)^2-(m+n)^2)?
Correct answer: A
On expansion, ((2m-n)^2=4m^2-4mn+n^2) and ((m+n)^2=m^2+2mn+n^2), so the difference is (3m^2-6mn). In exams, check the signs carefully.
View question detailsMathematics
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.
TOPIC PRACTICE
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Hard · Level 44 · polynomials,algebraic-identities,expansion,real-numbers
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- (,3m^2-6mn,)
- (,3m^2+2mn,)
- (,m^2-2mn,)
- (,3m^2-2n^2,)
Hard · Level 44 · polynomials,monomials,multiplication,negative-exponents
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- (,-12ab^2,)
- (,-12a^3b^2,)
- (,12ab^{-8},)
- (,-a b^2,)
Hard · Level 44 · polynomials,scientific-notation,rational-exponents,real-numbers
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- (,10^{-6},)
- (,10^{-4},)
- (,10^{-2},)
- (,10^{-8},)
Hard · Level 44 · polynomials,zero-exponent,negative-exponents,real-numbers
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- (,4,)
- (,2,)
- (,1,)
- (,\dfrac{1}{4},)
Hard · Level 44 · polynomials,rational-exponents,negative-exponents,fractions
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- (,\dfrac{4}{9},)
- (,\dfrac{9}{4},)
- (,\dfrac{2}{3},)
- (,\dfrac{8}{27},)
Hard · Level 44 · polynomials,surds,multiplication,real-numbers
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- (,-6,)
- (,6,)
- (,-2\sqrt{3},)
- (,2\sqrt{3},)
Hard · Level 44 · polynomials,cubes,algebraic-identities,expansion
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- (,x^3-8,)
- (,x^3+8,)
- (,x^3-4x-8,)
- (,x^3+4x-8,)
Easy · Level 44 · polynomials,substitution,evaluation,Operations on real numbers and the laws of exponents,Mathematics,Class 10 MCQ
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- -15
- -5
- 15
- -1
Hard · Level 44 · polynomials,rational-exponents,negative-exponents,algebra
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- (,\dfrac{a^2}{b},)
- (,\dfrac{a}{b^2},)
- (,a^2b,)
- (,\dfrac{b}{a^2},)
Hard · Level 44 · polynomials,powers-of-ten,common-factor,exponents
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- (,10,)
- (,9,)
- (,100,)
- (,1,)
Hard · Level 44 · polynomials,exponents,prime-factorisation,real-numbers
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- (,9,)
- (,6,)
- (,3,)
- (,36,)
Hard · Level 44 · polynomials,roots,surds,real-numbers
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- (,9,)
- (,8,)
- (,5,)
- (,4\sqrt{3},)
Hard · Level 44 · polynomials,exponents,equations,same-base
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- (,\dfrac{5}{2},)
- (,2,)
- (,\dfrac{3}{2},)
- (,4,)
Hard · Level 44 · polynomials,negative-exponents,division,algebraic-simplification
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- (,1+x,)
- (,1+\dfrac{1}{x},)
- (,x^{-5},)
- (,x,)
Hard · Level 44 · polynomials,negative-exponents,power-of-quotient,algebra
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- \(,\dfrac{1}{a^4b^6},\)
- \(,a^4b^6,\)
- \(,\dfrac{b^6}{a^4},\)
- \(,\dfrac{a^4}{b^6},\)
Hard · Level 44 · polynomials,surds,squares,real-numbers
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- (,18,)
- (,10,)
- (,12,)
- (,16,)
Hard · Level 44 · polynomials,algebraic-identities,simplification,binomials
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- (,24x,)
- (,12x,)
- (,36x^2,)
- (,8,)
Hard · Level 44 · polynomials,cubes,division,algebraic-identities
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- (,4x^2-2x+1,)
- (,4x^2+2x+1,)
- (,4x^2-1,)
- (,2x^2-2x+1,)
Hard · Level 44 · polynomials,powers-of-ten,negative-exponents,scientific-notation
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- (,450,)
- (,45,)
- (,0.00045,)
- (,4.5,)
Hard · Level 44 · polynomials,rational-exponents,fractions,roots
View options
- (,\dfrac{27}{8},)
- (,\dfrac{9}{8},)
- (,\dfrac{81}{16},)
- (,\dfrac{3}{2},)
What is the product of ((-4a^2b^{-3})(3a^{-1}b^5))?
Correct answer: A
The product of coefficients (-4) and (3) is (-12), and (a^{2-1}b^{-3+5}=ab^2). In exams, handle coefficients and exponents separately.
View question detailsWhat is the value of ((0.0001)^{\frac{3}{2}})?
Correct answer: A
Since (0.0001=10^{-4}), ((10^{-4})^{\frac{3}{2}}=10^{-6}). In exams, convert decimals into powers of (10).
View question detailsIf (x \neq 0), what is the value of (\dfrac{(5x^2)^0+x^0}{2^{-1}})?
Correct answer: A
Because ((5x^2)^0=1), (x^0=1), and (2^{-1}=\dfrac{1}{2}), the value is (4). In exams, apply the zero exponent rule only to a non-zero base.
View question detailsWhat is the value of \(\left(\dfrac{27}{8}\right)^{-\frac{2}{3}}\)?
Correct answer: A
\(\left(\dfrac{27}{8}\right)^{\frac{1}{3}}=\dfrac{3}{2}\), so \(\left(\dfrac{27}{8}\right)^{-\frac{2}{3}}=\left(\dfrac{3}{2}\right)^{-2}=\dfrac{4}{9}\). In exams, take the reciprocal for a negative exponent.
View question detailsSimplify: what is the value of (2\sqrt{3}(\sqrt{12}-\sqrt{27}))?
Correct answer: A
(\sqrt{12}=2\sqrt{3}) and (\sqrt{27}=3\sqrt{3}), so the inside value is (-\sqrt{3}) and the product is (-6). In exams, simplify the surds first.
View question detailsWhat is the expansion of ((x-2)(x^2+2x+4))?
Correct answer: A
This matches ((a-b)(a^2+ab+b^2)=a^3-b^3), so the answer is (x^3-8). In exams, identifying the identity makes expansion faster.
View question detailsFor the polynomial P(x)=2x^3-5x^2+x-7, what is the value of P(-1)?
Correct answer: A
The governing concept is evaluation of a polynomial by substitution. Replace every occurrence of x with -1, keeping brackets so that the signs and powers are handled correctly: P(-1)=2(-1)^3-5(-1)^2+(-1)-7. Since (-1)^3=-1 and (-1)^2=1, this becomes 2(-1)-5(1)-1-7=-2-5-1-7=-15. Thus option A is correct. Option B usually results from an arithmetic or sign error, option C reverses the final sign, and option D does not represent the value of the complete polynomial.
View question detailsIf (a>0) and (b>0), what is the simplified form of ((a^4b^{-2})^{\frac{1}{2}})?
Correct answer: A
Applying the outside exponent (\dfrac{1}{2}) gives (a^2b^{-1}=\dfrac{a^2}{b}). In exams, apply the fractional power to every factor.
View question detailsWhat is the value of (\dfrac{10^5-10^4}{9\times 10^3})?
Correct answer: A
Taking (10^4) common in the numerator gives (\dfrac{10^4(10-1)}{9\times 10^3}=10). In exams, taking a common factor makes calculation easier.
View question detailsWhat is the value of (\dfrac{6^4}{2^4 \times 3^2})?
Correct answer: A
Since (6^4=(2\times 3)^4=2^4\times 3^4), the value is (3^2=9). In exams, write a composite base in prime factors.
View question detailsWhat is the value of (\dfrac{\sqrt{48}}{\sqrt{3}}+\dfrac{\sqrt{75}}{\sqrt{3}})?
Correct answer: A
(\dfrac{\sqrt{48}}{\sqrt{3}}=\sqrt{16}=4) and (\dfrac{\sqrt{75}}{\sqrt{3}}=\sqrt{25}=5), so the sum is (9). In exams, simplify the division inside the root.
View question detailsIf (3^{2x-1}=81), what is the value of (x)?
Correct answer: A
Since (81=3^4), we get (2x-1=4) and (x=\dfrac{5}{2}). In exams, equate exponents when the bases are the same.
View question detailsIf (x \neq 0), what is the simplified form of (\dfrac{x^{-3}+x^{-2}}{x^{-3}})?
Correct answer: A
Dividing both terms by (x^{-3}) gives (1+x). In exams, divide each term separately by the denominator.
View question detailsIf \(a \neq 0\) and \(b \neq 0\), what is the simplified form of \(\left(\dfrac{a^2}{b^{-3}}\right)^{-2}\)?
Correct answer: A
Inside, \(\dfrac{a^2}{b^{-3}}=a^2b^3\), and applying the power (-2) gives \(\dfrac{1}{a^4b^6}\). In exams, simplify the inside part first.
View question detailsWhat is the value of ((\sqrt{2}+\sqrt{8})^2)?
Correct answer: A
Since (\sqrt{8}=2\sqrt{2}), ((\sqrt{2}+\sqrt{8})^2=(3\sqrt{2})^2=18). In exams, simplify the surd before squaring.
View question detailsWhat is the simplified form of ((3x+2)^2-(3x-2)^2)?
Correct answer: A
This is of the form ((A+B)^2-(A-B)^2=4AB), where (A=3x) and (B=2), so the answer is (24x). In exams, identities save time.
View question detailsWhat is the quotient when ((8x^3+1)) is divided by ((2x+1))?
Correct answer: A
Because (8x^3+1=(2x)^3+1^3=(2x+1)(4x^2-2x+1)). In exams, remember the identity for sum of cubes.
View question detailsIf (10^{-3}\times N=0.45), what is the value of (N)?
Correct answer: A
(N=\dfrac{0.45}{10^{-3}}=0.45\times 10^3=450). In exams, dividing by (10^{-3}) is like multiplying by (10^3).
View question detailsWhat is the value of \(\left(\dfrac{9}{4}\right)^{\frac{3}{2}}\)?
Correct answer: A
\(\left(\dfrac{9}{4}\right)^{\frac{1}{2}}=\dfrac{3}{2}\), so \(\left(\dfrac{9}{4}\right)^{\frac{3}{2}}=\left(\dfrac{3}{2}\right)^3=\dfrac{27}{8}\). In exams, take the square root first.
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