What 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.
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.
TOPIC PRACTICE
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Up to 25 questions from this page. Select your focus, then start.
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Hard · Level 2
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- (,3,)
- (,9,)
- (,\sqrt{40},)
- (,\dfrac{9}{5},)
Hard · Level 2
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- (,ab^2,)
- (,a^2b,)
- (,a^2b^2,)
- (,\dfrac{a}{b^2},)
Hard · Level 2
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- \(,x^4y^2,\)
- \(,\dfrac{x^4}{y^2},\)
- \(,x^2y^2,\)
- \(,x^4y^{-2},\)
Hard · Level 2
View options
- (,-30,)
- (,6,)
- (,-6,)
- (,30,)
Hard · Level 2
View options
- (,24,)
- (,8,)
- (,-24,)
- (,16,)
Hard · Level 2
View options
- (,-8,)
- (,8,)
- (,-\dfrac{1}{8},)
- (,\dfrac{1}{8},)
Hard · Level 2
View options
- (,\dfrac{x^2}{4},)
- (,4x^2,)
- (,\dfrac{4}{x^2},)
- (,\dfrac{1}{4x^2},)
Hard · Level 2
View options
- \(a^m\cdot a^n=a^{m+n}\)
- \((a^m)^n=a^{m+n}\)
- \(\dfrac{a^m}{a^n}=a^{mn}\)
- \((a+b)^m=a^m+b^m\)
Hard · Level 2
View options
- (,14,)
- (,8,)
- (,10,)
- (,4\sqrt{3},)
Hard · Level 2
View options
- (,x^2,)
- (,x^3,)
- (,x^{\frac{1}{2}},)
- (,x^6,)
Hard · Level 2
View options
- (,\dfrac{a^3}{b^4},)
- (,\dfrac{a}{b^2},)
- (,a^3b^4,)
- (,\dfrac{b^4}{a^3},)
Hard · Level 2
View options
- (,\dfrac{4}{9},)
- (,\dfrac{9}{4},)
- (,\dfrac{16}{81},)
- (,\dfrac{2}{3},)
Hard · Level 2
View options
- (,10^8,)
- (,10^4,)
- (,10^6,)
- (,10^{-8},)
Hard · Level 2
View options
- (,\dfrac{26}{5},)
- (,10,)
- (,\dfrac{24}{5},)
- (,\dfrac{6}{5},)
Hard · Level 2
View options
- (,x^2+4,)
- (,x^2-4,)
- (,x^2+16,)
- (,x^2-8,)
Hard · Level 2
View options
- (,2m^2+2n^2,)
- (,4mn,)
- (,m^2-n^2,)
- (,2m^2-2n^2,)
Hard · Level 2
View options
- (,4x^2+3x-3,)
- (,4x^2-7x+5,)
- (,2x^2+3x-3,)
- (,4x^2+7x-5,)
Hard · Level 2
View options
- (,3x^3-5x+12,)
- (,3x^3+x+2,)
- (,7x^3+x+2,)
- (,3x^3-5x-2,)
Hard · Level 2
View options
- (,6x^2+5x-4,)
- (,6x^2-5x-4,)
- (,6x^2+11x-4,)
- (,6x^2+5x+4,)
Hard · Level 2
View options
- (,\dfrac{x^6}{y^5},)
- (,\dfrac{x^4}{y},)
- (,x^6y^5,)
- (,\dfrac{y^5}{x^6},)
Hard · Level 2
View options
- \(,\dfrac{a^3}{b^4},\)
- \(,\dfrac{b^4}{a^3},\)
- \(,a^3b^4,\)
- \(,\dfrac{1}{a^3b^4},\)
Hard · Level 2
View options
- (,\dfrac{1}{16},)
- (,16,)
- (,\dfrac{1}{8},)
- (,8,)
Hard · Level 2
View options
- (,\dfrac{6}{5},)
- (,\dfrac{5}{6},)
- (,5,)
- (,\dfrac{1}{5},)
Hard · Level 2
View options
- (,243,)
- (,81,)
- (,27,)
- (,\dfrac{1}{243},)
Hard · Level 2
View options
- (,\dfrac{b^5}{a^3},)
- (,\dfrac{a^3}{b^5},)
- (,a^{-5}b^7,)
- (,\dfrac{b^7}{a^5},)
If (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.
If \(y \neq 0\), 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.
If (a=2) and (b=-3), what is the value of (a^2b-ab^2)?
Correct answer: A
The direct answer is option A, -30. Substitute a=2 and b=-3 carefully into a^2b-ab^2. First, a^2=2^2=4. Next, b^2=(-3)^2=9; the square is positive because two negative factors make a positive product. Therefore a^2b=4(-3)=-12 and ab^2=2(9)=18. The expression is a^2b-ab^2=-12-18=-30. Option A is correct. Option B, 6, does not follow from the substitution and usually comes from mishandling signs or powers. Option C, -6, is also incorrect because it does not equal the two calculated terms combined. Option D, 30, has the wrong sign; the second term is subtracted, so -12-18 remains negative. A reliable method is to calculate powers first, then multiplication, and finally the subtraction. The supplied answer is consistent.
What 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.
What 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.
If (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.
If \(a\) 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.
What 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.
What 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.
If (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.
What 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.
What 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.
If (a^2=5) and (a>0), what is the value of (a^{-2}+a^2)?
Correct answer: A
Because (a^{-2}=\dfrac{1}{a^2}=\dfrac{1}{5}), (\dfrac{1}{5}+5=\dfrac{26}{5}). In exams, write (a^{-2}) as (\dfrac{1}{a^2}).
If (x^2 \neq 4), what is the simplified form of (\dfrac{x^4-16}{x^2-4})?
Correct answer: A
(x^4-16=(x^2-4)(x^2+4)), so the simplified form is (x^2+4). In exams, treat (x^4) as ((x^2)^2) for factorisation.
What is the simplified form of ((m+n)^2+(m-n)^2)?
Correct answer: A
When both expansions are added, (2mn) and (-2mn) cancel, giving (2m^2+2n^2). In exams, notice opposite middle terms.
What is the sum of ((3x^2-2x+1)+(x^2+5x-4))?
Correct answer: A
Adding like terms gives (3x^2+x^2=4x^2), (-2x+5x=3x), and (1-4=-3). In exams, add like terms in columns.
What is the simplified form of ((5x^3-2x+7)-(2x^3+3x-5))?
Correct answer: A
Changing the signs of the second bracket gives (5x^3-2x+7-2x^3-3x+5), so the answer is (3x^3-5x+12). In exams, change the sign of every term in the bracket during subtraction.
What is the expansion of ((2x-1)(3x+4))?
Correct answer: A
The direct answer is A: 6x²+5x−4. Multiply every term in the first bracket by every term in the second bracket. First, 2x×3x=6x². Next, 2x×4=8x. Then, −1×3x=−3x. Finally, −1×4=−4. Combining like terms gives 6x²+8x−3x−4=6x²+5x−4. Option A is correct. Option B has the wrong sign for the x-term; the middle terms actually combine as 8x−3x=5x. Option C adds the cross terms as 8x+3x and therefore gets 11x, ignoring the negative sign. Option D has the wrong constant sign; (−1)(4)=−4, not +4. A useful check is that the leading coefficient is 2×3=6 and the constant is (−1)×4=−4. Memory cue: use FOIL carefully and preserve every sign.
If (x \neq 0) and (y \neq 0), what is the simplified form of (\dfrac{x^5y^{-2}}{x^{-1}y^3})?
Correct answer: A
(x^{5-(-1)}=x^6) and (y^{-2-3}=y^{-5}), so the form is (\dfrac{x^6}{y^5}). In exams, simplify the exponent of each variable separately.
If \(a \neq 0\) and \(b \neq 0\), what is the simplified form of \(\left(\dfrac{a^{-2}b}{ab^{-3}}\right)^{-1}\)?
Correct answer: A
The expression inside is \(a^{-3}b^4\), and the power (-1) gives its reciprocal \(\dfrac{a^3}{b^4}\). In exams, apply the outer negative power at the end.
What is the value of ((\sqrt[3]{64})^{-2})?
Correct answer: A
Since (\sqrt[3]{64}=4), (4^{-2}=\dfrac{1}{16}). In exams, first evaluate the root and then apply the negative exponent.
What is the value of (\dfrac{1}{2^{-1}+3^{-1}})?
Correct answer: A
(2^{-1}+3^{-1}=\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}), so the whole value is (\dfrac{6}{5}). In exams, simplify the denominator first.
Simplify: what is the value of (\dfrac{3^4 \times 9^{-1}}{27^{-1}})?
Correct answer: A
Here (9^{-1}=3^{-2}) and (27^{-1}=3^{-3}), so the value is (3^{4-2-(-3)}=3^5=243). In exams, convert all terms to the same base.
If (a \neq 0) and (b \neq 0), what is the simplified form of (\dfrac{(a^{-2}b^3)^2}{(ab^{-1})^{-1}})?
Correct answer: A
The numerator is ((a^{-2}b^3)^2=a^{-4}b^6) and the denominator is ((ab^{-1})^{-1}=a^{-1}b), so the answer is (\dfrac{b^5}{a^3}). In exams, apply the outside power first.
QUIZ COMPLETE