What is the value of ((2^{-3}+2^{-2})^{-1})?
Correct answer: A
Inside, (2^{-3}+2^{-2}=\dfrac{1}{8}+\dfrac{1}{4}=\dfrac{3}{8}), so the power (-1) gives (\dfrac{8}{3}). In exams, simplify the bracket first.
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,negative-exponents,fractions,real-numbers
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- (,\dfrac{8}{3},)
- (,\dfrac{3}{8},)
- (,8,)
- (,\dfrac{1}{3},)
Hard · Level 44 · polynomials,factorisation,difference-of-squares,simplification
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- (,x^2-9,)
- (,x^2+9,)
- (,x^2-81,)
- (,x^2+81,)
Hard · Level 44 · polynomials,addition,like-terms,operations
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- (,3x^2+4x-3,)
- (,5x^2+4x-3,)
- (,3x^2-10x+15,)
- (,3x^2+10x-3,)
Hard · Level 44 · polynomials,subtraction,like-terms,operations
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- (,-3y^3-5y+13,)
- (,-3y^3+3y-3,)
- (,7y^3+3y-3,)
- (,-3y^3-5y-13,)
Hard · Level 44 · polynomials,cubes,expansion,algebraic-identities
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- (,6x^2+2,)
- (,6x^2-2,)
- (,2x^3+2,)
- (,3x^2+1,)
Hard · Level 44 · polynomials,laws of exponents,real numbers,powers,algebraic reasoning
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- 28
- 14
- 49
- 21
Hard · Level 44 · polynomials,rational-exponents,negative-exponents,algebra
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- (,\dfrac{4x^2}{y},)
- (,4x^2y,)
- (,\dfrac{8x^3}{y},)
- (,\dfrac{4x^3}{y^2},)
Hard · Level 44 · polynomials,roots,exponents,algebraic-simplification
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- (,a^2b,)
- (,a^2b^2,)
- (,ab^2,)
- (,a^4b,)
Hard · Level 44 · polynomials,cubes,algebraic-identities,simplification
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- (,3x^2+y^2,)
- (,x^2+3y^2,)
- (,3x^2-y^2,)
- (,x^2-y^2,)
Hard · Level 44 · polynomials,negative-exponents,same-base,exponents
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- (,243,)
- (,27,)
- (,81,)
- (,\dfrac{1}{243},)
Hard · Level 44 · polynomials,exponents,equations,division-law
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- (,3,)
- (,2,)
- (,5,)
- (,7,)
Hard · Level 44 · polynomials,monomials,negative-exponents,division
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- (,2x^2,)
- (,2x,)
- (,\dfrac{2}{x^2},)
- (,6x^2,)
Hard · Level 44 · polynomials,surds,simplification,real-numbers
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- (,8\sqrt{2},)
- (,6\sqrt{2},)
- (,10\sqrt{2},)
- (,\sqrt{120},)
Hard · Level 44 · polynomials,rationalisation,surds,conjugate
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- (,6+3\sqrt{3},)
- (,6-3\sqrt{3},)
- (,2+\sqrt{3},)
- (,\dfrac{3}{2+\sqrt{3}},)
Hard · Level 44 · polynomials,negative-exponents,fractions,real-numbers
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- (,20,)
- (,\dfrac{1}{20},)
- (,9,)
- (,\dfrac{20}{9},)
Hard · Level 44 · polynomials,factorisation,division,simplification
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- (,x+2,)
- (,x+1,)
- (,x+3,)
- (,x-2,)
Hard · Level 44 · polynomials,rational-exponents,roots,real-numbers
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- (,5,)
- (,25,)
- (,\dfrac{1}{5},)
- (,125,)
Hard · Level 44 · polynomials,roots,real-numbers,equations
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- (,63,)
- (,21,)
- (,42,)
- (,49,)
Hard · Level 44 · polynomials,negative-exponents,fractions,algebra
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- (,a+b,)
- (,\dfrac{a+b}{ab},)
- (,\dfrac{1}{a+b},)
- (,ab(a+b),)
Hard · Level 44 · polynomials,rational-exponents,power-of-power,real-numbers
View options
- (,12,)
- (,24,)
- (,6,)
- (,36,)
If (x^2+9 \neq 0), what is the simplified form of (\dfrac{x^4-81}{x^2+9})?
Correct answer: A
(x^4-81=(x^2-9)(x^2+9)), so the simplified form is (x^2-9). In exams, treat (x^4) as ((x^2)^2) while factoring.
View question detailsWhat is the sum of ((4x^2-3x+6)+(-x^2+7x-9))?
Correct answer: A
Adding like terms gives (4x^2-x^2=3x^2), (-3x+7x=4x), and (6-9=-3). In exams, add like terms separately.
View question detailsWhat is the simplified form of ((2y^3-y+5)-(5y^3+4y-8))?
Correct answer: A
Changing all signs in the second bracket gives (2y^3-y+5-5y^3-4y+8). In exams, change the sign of every term during subtraction.
View question detailsWhat is the simplified form of ((x+1)^3-(x-1)^3)?
Correct answer: A
On expansion, ((x+1)^3=x^3+3x^2+3x+1) and ((x-1)^3=x^3-3x^2+3x-1), so the difference is (6x^2+2). In exams, expand cubes carefully.
View question detailsIf \(a^m=2\) and \(a^n=7\), what is the value of \(a^{2m+n}\)?
Correct answer: A
Using the laws of exponents, \(a^{2m+n}=a^{2m}\cdot a^n=(a^m)^2\cdot a^n\). Substituting the given values gives \(2^2\times 7=4\times 7=28\). Therefore, the correct answer is 28. In an exam, split a sum in the exponent into a product of powers; multiplying \(a^m\) and \(a^n\) directly would give 14, which corresponds to \(a^{m+n}\).
View question detailsIf (y \neq 0), what is the simplified form of ((64x^6y^{-3})^{\frac{1}{3}})?
Correct answer: A
((64)^{\frac{1}{3}}=4), ((x^6)^{\frac{1}{3}}=x^2), and ((y^{-3})^{\frac{1}{3}}=y^{-1}), so the answer is (\dfrac{4x^2}{y}). In exams, apply the exponent to each factor.
View question detailsIf (a>0) and (b>0), what is the simplified form of (\sqrt{a^4b^2})?
Correct answer: A
Because (\sqrt{a^4}=a^2) and (\sqrt{b^2}=b), the simplified form is (a^2b). In exams, note the positive condition.
View question detailsIf (y \neq 0), what is the simplified form of (\dfrac{(x+y)^3-(x-y)^3}{2y})?
Correct answer: A
The numerator difference is (6x^2y+2y^3=2y(3x^2+y^2)), so division gives (3x^2+y^2). In exams, take out the common factor.
View question detailsWhat is the value of (\dfrac{3^{-2}\times 9^2}{27^{-1}})?
Correct answer: A
(9^2=3^4) and (27^{-1}=3^{-3}), so the value is (3^{-2+4-(-3)}=3^5=243). In exams, be careful while subtracting a negative exponent.
View question detailsIf (a \neq 0), (a \neq 1), and (\dfrac{a^5}{a^k}=a^2), what is the value of (k)?
Correct answer: A
(\dfrac{a^5}{a^k}=a^{5-k}), so (5-k=2) and (k=3). In exams, subtract exponents using the division law.
View question detailsIf (x \neq 0), what is the simplified form of (\dfrac{(2x)^3(3x^{-2})}{12x^{-1}})?
Correct answer: A
The numerator is ((2x)^3(3x^{-2})=8x^3\cdot 3x^{-2}=24x), and (\dfrac{24x}{12x^{-1}}=2x^2). In exams, simplify both coefficient and variable parts.
View question detailsWhat is the simplified form of (\sqrt{98}+\sqrt{72}-\sqrt{50})?
Correct answer: A
(\sqrt{98}=7\sqrt{2}), (\sqrt{72}=6\sqrt{2}), and (\sqrt{50}=5\sqrt{2}), so the answer is (8\sqrt{2}). In exams, first write all surds in simplest form.
View question detailsWhich form is obtained by rationalising the denominator of (\dfrac{3}{2-\sqrt{3}})?
Correct answer: A
Multiplying by (2+\sqrt{3}) makes the denominator (4-3=1). In exams, multiply both numerator and denominator by the conjugate.
View question detailsWhat is the value of (\dfrac{1}{4^{-1}-5^{-1}})?
Correct answer: A
(4^{-1}-5^{-1}=\dfrac{1}{4}-\dfrac{1}{5}=\dfrac{1}{20}), so the whole value is (20). In exams, first convert negative powers into fractions.
View question detailsIf (x+1 \neq 0), what is the simplified form of (\dfrac{x^2+3x+2}{x+1})?
Correct answer: A
Because (x^2+3x+2=(x+1)(x+2)), the simplified form is (x+2). In exams, factorise trinomials carefully.
View question detailsWhat is the value of (\dfrac{125^{\frac{2}{3}}}{25^{\frac{1}{2}}})?
Correct answer: A
(125^{\frac{2}{3}}=25) and (25^{\frac{1}{2}}=5), so the value is (5). In exams, separate fractional exponents into root and power.
View question detailsIf (\sqrt{n}=3\sqrt{7}), what is the value of (n)?
Correct answer: A
Squaring both sides gives (n=(3\sqrt{7})^2=9\times 7=63). In exams, square both sides in a square root equation.
View question detailsIf (a \neq 0) and (b \neq 0), what is the simplified form of (\dfrac{a^{-1}+b^{-1}}{(ab)^{-1}})?
Correct answer: A
The numerator is (a^{-1}+b^{-1}=\dfrac{a+b}{ab}) and the denominator is ((ab)^{-1}=\dfrac{1}{ab}), so the answer is (a+b). In exams, make a common denominator.
View question detailsWhat is the value of ((2^5)^{\frac{2}{5}}\times (3^3)^{\frac{1}{3}})?
Correct answer: A
((2^5)^{\frac{2}{5}}=2^2=4) and ((3^3)^{\frac{1}{3}}=3), so the product is (12). In exams, apply the power of a power law.
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