If (5^n=\dfrac{1}{125}), what is the value of (n)?
Correct answer: A
Since (125=5^3), (\dfrac{1}{125}=5^{-3}), so (n=-3). In exams, connect a reciprocal with a negative exponent.
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 43 · polynomials,negative-exponents,equations,same-base
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- (,-3,)
- (,3,)
- (,-2,)
- (,\dfrac{1}{3},)
Hard · Level 43 · polynomials,exponents,common-factor,real-numbers
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- (,6,)
- (,7,)
- (,1,)
- (,49,)
Hard · Level 43 · polynomials,exponents,addition,same-base
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- (,4,)
- (,2,)
- (,8,)
- (,16,)
Hard · Level 43 · polynomials,multiplication,expansion,real-numbers
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- (,x^2-x-6,)
- (,x^2+x-6,)
- (,x^2-x+6,)
- (,x^2-5x-6,)
Hard · Level 43 · polynomials,division,cubes,algebraic-identities
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- (,x^2+2x+4,)
- (,x^2-2x+4,)
- (,x^2+4x+2,)
- (,x^2-4,)
Hard · Level 43 · polynomials,polynomial evaluation,substitution,exponents
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- 1
- 0
- 2
- -1
Hard · Level 43 · polynomials,real-numbers,roots,surds
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- (,18,)
- (,36,)
- (,9,)
- (,6\sqrt{3},)
Hard · Level 43 · polynomials,real-numbers,roots,simplification
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- (,3,)
- (,9,)
- (,\sqrt{40},)
- (,\dfrac{9}{5},)
Hard · Level 43 · polynomials,roots,exponents,algebraic-simplification
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- (,ab^2,)
- (,a^2b,)
- (,a^2b^2,)
- (,\dfrac{a}{b^2},)
Hard · Level 43 · polynomials,negative-exponents,power-of-quotient,algebra
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- \(,x^4y^2,\)
- \(,\dfrac{x^4}{y^2},\)
- \(,x^2y^2,\)
- \(,x^4y^{-2},\)
Hard · Level 43 · polynomials,evaluation,real-numbers,substitution
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- (,-30,)
- (,6,)
- (,-6,)
- (,30,)
Hard · Level 43 · polynomials,real-numbers,integer-powers,signs
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- (,24,)
- (,8,)
- (,-24,)
- (,16,)
Hard · Level 43 · polynomials,negative-exponents,fractions,signs
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- (,-8,)
- (,8,)
- (,-\dfrac{1}{8},)
- (,\dfrac{1}{8},)
Hard · Level 43 · polynomials,negative-exponents,power-of-product,algebra
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- (,\dfrac{x^2}{4},)
- (,4x^2,)
- (,\dfrac{4}{x^2},)
- (,\dfrac{1}{4x^2},)
Hard · Level 43 · real numbers, laws of exponents, integer exponents, algebraic identities, polynomials
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- \(a^m\cdot a^n=a^{m+n}\)
- \((a^m)^n=a^{m+n}\)
- \(\dfrac{a^m}{a^n}=a^{mn}\)
- \((a+b)^m=a^m+b^m\)
Hard · Level 43 · polynomials,surds,real-numbers,identities
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- (,14,)
- (,8,)
- (,10,)
- (,4\sqrt{3},)
Hard · Level 43 · polynomials,rational-exponents,power-of-power,algebra
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- (,x^2,)
- (,x^3,)
- (,x^{\frac{1}{2}},)
- (,x^6,)
Hard · Level 43 · polynomials,negative-exponents,division,algebra
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- (,\dfrac{a^3}{b^4},)
- (,\dfrac{a}{b^2},)
- (,a^3b^4,)
- (,\dfrac{b^4}{a^3},)
Hard · Level 43 · polynomials,rational-exponents,negative-exponents,fractions
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- (,\dfrac{4}{9},)
- (,\dfrac{9}{4},)
- (,\dfrac{16}{81},)
- (,\dfrac{2}{3},)
Hard · Level 43 · polynomials,powers-of-ten,negative-exponents,division
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- (,10^8,)
- (,10^4,)
- (,10^6,)
- (,10^{-8},)
What is the value of (\dfrac{7^5-7^4}{7^4})?
Correct answer: A
Taking (7^4) common in the numerator gives (\dfrac{7^4(7-1)}{7^4}=6). In exams, taking a common factor makes calculation shorter.
View question detailsWhat is the value of (\dfrac{2^{10}+2^{10}}{2^9})?
Correct answer: A
The numerator is (2^{10}+2^{10}=2\times 2^{10}=2^{11}), so (\dfrac{2^{11}}{2^9}=2^2=4). In exams, first combine like terms and then apply exponent laws.
View question detailsWhat is the expansion of ((x+2)(x-3))?
Correct answer: A
Using the distributive law, (x(x-3)+2(x-3)=x^2-x-6). In exams, check the sign of the middle term carefully.
View question detailsWhat is the quotient when ((x^3-8)) is divided by ((x-2))?
Correct answer: A
Because (x^3-8=(x-2)(x^2+2x+4)), the quotient is (x^2+2x+4). In exams, remember the identity for cubes.
View question detailsFor the polynomial \\(P(x)=x^3-4x+1\\), what is the value of \\(P(2)\\) when \\(x=2\\) is substituted?
Correct answer: A
Substituting \(x=2\), we get \(P(2)=2^3-4(2)+1=8-8+1=1\). Hence, option A is correct. Option B results from forgetting the final \(+1\). Exam tip: substitute the value carefully and simplify each term, especially the term containing the exponent.
View question detailsWhat is the value of (\sqrt{12}\times \sqrt{27})?
Correct answer: A
(\sqrt{12}\times \sqrt{27}=\sqrt{324}=18). In exams, use (\sqrt{a}\sqrt{b}=\sqrt{ab}) for non-negative numbers.
View question detailsWhat is the simplified value of (\dfrac{\sqrt{45}}{\sqrt{5}})?
Correct answer: A
(\dfrac{\sqrt{45}}{\sqrt{5}}=\sqrt{\dfrac{45}{5}}=\sqrt{9}=3). In exams, simplify division inside the root first.
View question detailsIf (a>0) and (b>0), what is the simplified form of (\sqrt{a^2b^4})?
Correct answer: A
Because (\sqrt{a^2}=a) and (\sqrt{b^4}=b^2), the answer is (ab^2). In exams, note the condition that variables are positive.
View question detailsIf \(y \neq 0\), what is the simplified form of \(\left(\dfrac{x^2}{y^{-1}}\right)^2\)?
Correct answer: A
\(\dfrac{x^2}{y^{-1}}=x^2y\), so the whole square is \(x^4y^2\). In exams, simplify a negative exponent by moving its position.
View question detailsIf (a=2) and (b=-3), what is the value of (a^2b-ab^2)?
Correct answer: A
(a^2b-ab^2=4(-3)-2(9)=-12-18=-30). In exams, the square of a negative number is always positive.
View question detailsWhat is the value of ((-2)^4-(-2)^3)?
Correct answer: A
((-2)^4=16) and ((-2)^3=-8), so (16-(-8)=24). In exams, odd and even powers have different signs.
View question detailsWhat is the value of \(\left(-\dfrac{1}{2}\right)^{-3}\)?
Correct answer: A
A negative exponent inverts the fraction, so \(\left(-\dfrac{1}{2}\right)^{-3}=(-2)^3=-8\). In exams, keep the sign of a negative base according to the power.
View question detailsIf (x \neq 0), what is the simplified form of ((4x^{-2})^{-1})?
Correct answer: A
((4x^{-2})^{-1}=4^{-1}x^2=\dfrac{x^2}{4}). In exams, apply the outside exponent to every factor of a product.
View question detailsIf \(a\) and \(b\) are non-zero real numbers and \(m,n\) are integers, which of the following law of exponents is always valid?
Correct answer: A
When powers with the same base are multiplied, their exponents are added, so \(a^m\cdot a^n=a^{m+n}\). In option B, exponents multiply: \((a^m)^n=a^{mn}\). Exam tip: first check whether the bases are the same before applying an exponent law.
View question detailsWhat is the value of ((2+\sqrt{3})^2+(2-\sqrt{3})^2)?
Correct answer: A
When the two squares are added, the surd terms cancel and (7+7=14). In exams, irrational terms often cancel in conjugate expressions.
View question detailsWhat is the simplified form of ((x^{\frac{1}{3}})^6)?
Correct answer: A
By the power of a power law, ((x^{\frac{1}{3}})^6=x^{\frac{6}{3}}=x^2). In exams, multiply the exponents.
View question detailsIf (a \neq 0) and (b \neq 0), what is the simplified form of (a^2b^{-3}\div a^{-1}b)?
Correct answer: A
In division, (a^{2-(-1)}=a^3) and (b^{-3-1}=b^{-4}), so the answer is (\dfrac{a^3}{b^4}). In exams, subtract exponents of like variables separately.
View question detailsWhat is the value of \(\left(\dfrac{81}{16}\right)^{-\frac{1}{2}}\)?
Correct answer: A
First, \(\left(\dfrac{81}{16}\right)^{\frac{1}{2}}=\dfrac{9}{4}\), then the negative exponent gives (\dfrac{4}{9}). In exams, check both the square root and the reciprocal.
View question detailsWhat is the simplified form of (\dfrac{(10^3)^2}{10^{-2}})?
Correct answer: A
((10^3)^2=10^6) and (\dfrac{10^6}{10^{-2}}=10^{6-(-2)}=10^8). In exams, be careful while subtracting a negative exponent.
View question detailsQUIZ COMPLETE