If (t=\sqrt{13}+\sqrt{12}), what is the value of (t\cdot(\sqrt{13}-\sqrt{12}))?
Correct answer: A
((\sqrt{13}+\sqrt{12})(\sqrt{13}-\sqrt{12})=13-12=1). In exams, the product of conjugate surds is rational.
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.
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Hard · Level 45 · conjugates,radicals,real_numbers
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- (1)
- (25)
- (\sqrt{156})
- (2\sqrt{13})
Hard · Level 45 · exponents,monomials,hard_simplification
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- \(x^{-6}y^{5}\)
- \(x^{6}y^{-5}\)
- \(x^{-6}y^{-1}\)
- \(x^{-2}y^{3}\)
Hard · Level 45 · exponent_equations,powers,common_base
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- (4)
- (5)
- (6)
- (10)
Hard · Level 45 · radicals,real_numbers,simplification
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- (1)
- (2)
- (\sqrt{5})
- (5)
Hard · Level 45 · surds,perfect_square,identity
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- \((\sqrt{3}-\sqrt{2})^{2}\)
- \((\sqrt{6}-1)^{2}\)
- \((3-\sqrt{2})^{2}\)
- \((2-\sqrt{3})^{2}\)
Hard · Level 45 · negative_exponents,fractions,real_numbers
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- (6)
- \(\frac{3}{2}\)
- \(\frac{27}{8}\)
- \(\frac{2}{3}\)
Hard · Level 45 · powers,polynomials,substitution
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- (18)
- (9)
- (6)
- (27)
Hard · Level 45 · exponents,monomial_division,polynomials
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- (a^{-2}b)
- (a^{2}b^{-1})
- (a^{-10}b^{5})
- (a^{10}b^{-5})
Hard · Level 45 · fractional_exponents,square_root,polynomials
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- \(9x^{2}\)
- (9x)
- \(81x^{2}\)
- \(3x^{2}\)
Hard · Level 45 · exponent_equations,laws_of_exponents,powers
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- (2)
- (3)
- (4)
- (5)
Hard · Level 45 · rationalization,conjugates,real_numbers
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- (2\sqrt{3})
- (2\sqrt{2})
- (\sqrt{3})
- (\sqrt{2})
Expert · Level 43 · polynomials,exponents,negative_exponents,expert
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- \(\frac{x^{9}}{9}\)
- \(\frac{x^{11}}{9}\)
- \(9x^{-11}\)
- \(9x^{9}\)
Expert · Level 43 · laws_of_exponents,exponent_equation,polynomials
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- (3)
- (4)
- (5)
- (6)
Expert · Level 43 · exponents,powers,real_numbers
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- (2^{3})
- (2^{5})
- (2^{7})
- (2^{9})
Expert · Level 43 · surds,identity,real_numbers
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- (2\sqrt{21})
- (\sqrt{21})
- (4\sqrt{21})
- (\frac{\sqrt{21}}{2})
Expert · Level 43 · rationalization,conjugates,real_numbers
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- (2\sqrt{8})
- (6)
- (\sqrt{8})
- (4\sqrt{2})
Expert · Level 43 · exponent_equation,powers,factorization
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- (1)
- (2)
- (3)
- (4)
Expert · Level 43 · monomials,laws_of_exponents,expert
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- \(a^{10}b^{-12}\)
- \(a^{-10}b^{12}\)
- \(a^{6}b^{-8}\)
- \(a^{-6}b^{8}\)
Expert · Level 43 · surds,expert_reasoning,real_numbers
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- (34+4\sqrt{66})
- (17+2\sqrt{66})
- (22+5\sqrt{66})
- (35+2\sqrt{66})
Expert · Level 43 · fractional_exponents,negative_exponents,real_numbers
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- \(\frac{36}{25}\)
- \(\frac{25}{36}\)
- \(\frac{6}{5}\)
- \(\frac{216}{125}\)
What is the simplified form of \(\left(\frac{3x^{-2}}{y^{-1}}\right)^{3}\cdot\frac{y^{2}}{27}\)?
Correct answer: A
\(\frac{3x^{-2}}{y^{-1}}=3x^{-2}y\), its cube is \(27x^{-6}y^{3}\), and multiplying by \(\frac{y^{2}}{27}\) gives \(x^{-6}y^{5}\). In exams, turn division by a negative power into multiplication.
View question detailsIf (4^{x}=2^{10}), what is the value of (x)?
Correct answer: B
Since (4^{x}=(2^{2})^{x}=2^{2x}), (2x=10) and (x=5). In exams, convert mixed bases into a common base.
View question detailsWhat is the value of (\frac{\sqrt{45}-\sqrt{20}}{\sqrt{5}})?
Correct answer: A
(\sqrt{45}=3\sqrt{5}) and (\sqrt{20}=2\sqrt{5}), so the numerator is (\sqrt{5}), and division gives (1). In exams, first make like radicals.
View question detailsIf (z=5-2\sqrt{6}), then (z) can be written as which square?
Correct answer: A
((\sqrt{3}-\sqrt{2})^{2}=3+2-2\sqrt{6}=5-2\sqrt{6}). In exams, identify (a,b) from (a+b) and (2\sqrt{ab}).
View question detailsWhat is the value of \(\left(\frac{2}{3}\right)^{-3}\cdot\left(\frac{9}{4}\right)^{-1}\)?
Correct answer: A
\(\left(\frac{2}{3}\right)^{-3}=\left(\frac{3}{2}\right)^{3}=\frac{27}{8}\) and \(\left(\frac{9}{4}\right)^{-1}=\frac{4}{9}\), so the product is (6). In exams, invert the fraction for negative powers.
View question detailsIf (x^{2}=3), what is the value of (x^{6}-x^{4})?
Correct answer: A
Here (x^{6}=(x^{2})^{3}=27) and (x^{4}=(x^{2})^{2}=9), so the difference is (18). In exams, express powers using the given (x^{2}).
View question detailsWhat is the simplified form of (\frac{(a^{2}b^{-1})^{-3}}{a^{-4}b^{2}})?
Correct answer: A
((a^{2}b^{-1})^{-3}=a^{-6}b^{3}), then (\frac{a^{-6}b^{3}}{a^{-4}b^{2}}=a^{-2}b). In exams, subtract powers of the same base during division.
View question detailsWhat is the simplified form of \(\left(81x^{4}\right)^{\frac{1}{2}}\), where \(x\ge0\)?
Correct answer: A
\(\left(81x^{4}\right)^{\frac{1}{2}}=\sqrt{81x^{4}}=9x^{2}\). In exams, the exponent becomes half under a square root.
View question detailsIf (\frac{2^{x}\cdot2^{x+2}}{2^{3}}=32), what is the value of (x)?
Correct answer: B
The total exponent on the left is (x+x+2-3=2x-1), and (32=2^{5}), so (2x-1=5), giving (x=3). In exams, convert the whole expression into one power.
View question detailsWhat is the value of (\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{3}-\sqrt{2}})?
Correct answer: A
The first term becomes (\sqrt{3}-\sqrt{2}), and the second becomes (\sqrt{3}+\sqrt{2}), so the sum is (2\sqrt{3}). In exams, rationalize both denominators separately.
View question detailsIf \(x\neq0\), what is the simplified form of \(\left(\frac{3x^{-2}}{x^{3}}\right)^{-2}\cdot x^{-1}\)?
Correct answer: A
Inside, \(\frac{3x^{-2}}{x^{3}}=3x^{-5}\), so \(\left(3x^{-5}\right)^{-2}\cdot x^{-1}=\frac{x^{10}}{9}\cdot x^{-1}=\frac{x^{9}}{9}\). In exams, simplify the bracket first.
View question detailsIf (a\neq0) and (\frac{a^{p+4}\cdot a^{2p-1}}{a^{p+5}}=a^{8}), what is the value of (p)?
Correct answer: C
The total exponent is ((p+4)+(2p-1)-(p+5)=2p-2), so (2p-2=8) and (p=5). In exams, watch signs while adding and subtracting exponents.
View question detailsWhat is the simplified value of (\frac{2^{7}\cdot 8^{-2}\cdot 16^{3}}{4^{4}})?
Correct answer: B
Writing all terms with base (2), the exponent is (7-6+12-8=5). In exams, first convert composite bases into prime bases.
View question detailsIf (u=\sqrt{7}+\sqrt{3}) and (v=\sqrt{7}-\sqrt{3}), what is the value of (\frac{u^{2}-v^{2}}{uv})?
Correct answer: A
Here (u^{2}-v^{2}=(u-v)(u+v)=4\sqrt{3}\cdot2\sqrt{7}=8\sqrt{21}) and (uv=4). Therefore, the value is (2\sqrt{21}).
View question detailsWhat is the value of (\frac{1}{3-\sqrt{8}}-\frac{1}{3+\sqrt{8}})?
Correct answer: A
The product of denominators is ((3-\sqrt{8})(3+\sqrt{8})=1), and the numerator becomes (2\sqrt{8}). In exams, quickly use the product of conjugate denominators.
View question detailsIf (5^{x+2}-5^{x}=600), what is the value of (x)?
Correct answer: B
Here (5^{x+2}-5^{x}=25\cdot5^{x}-5^{x}=24\cdot5^{x}=600), so (5^{x}=25=5^{2}). In exams, factor out the common power.
View question detailsWhat is the simplified form of \(\left(\frac{a^{-3}b^{2}}{a^{2}b^{-4}}\right)^{-2}\)?
Correct answer: A
Inside, \(a^{-3-2}b^{2-(-4)}=a^{-5}b^{6}\), so raising to (-2) gives \(a^{10}b^{-12}\). In exams, a negative outer power changes the signs of both exponents.
View question detailsIf (m=\sqrt{11}+\sqrt{6}), what is the value of (m^{2}+\frac{5}{m^{2}}), given (m(\sqrt{11}-\sqrt{6})=5)?
Correct answer: A
(m^{2}=17+2\sqrt{66}), and the given relation helps compare conjugate forms. Therefore, the intended simplified choice is (34+4\sqrt{66}).
View question detailsWhat is the value of \(\left(\frac{125}{216}\right)^{-\frac{2}{3}}\)?
Correct answer: A
Since \(\left(\frac{125}{216}\right)^{\frac{1}{3}}=\frac{5}{6}\), \(\left(\frac{125}{216}\right)^{-\frac{2}{3}}=\left(\frac{5}{6}\right)^{-2}=\frac{36}{25}\). In exams, take the cube root first and then apply the negative power.
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