Practice
Laws of Exponents and Scientific Notation Practice
Fifty original questions on exponent structures, base conditions, reciprocals, symbolic powers, scientific-notation conversion, operations, normalization, and error analysis.
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Question 1
Explanation
Multiply the coefficients: \(3(4)=12\). The bases match, so add exponents: \(2+5=7\). The product is \(12x^{7}\).
Question 2
Explanation
Multiply the coefficients: \(-2(5)=-10\). The bases match, so add exponents: \(4+3=7\). The product is \(-10a^{7}\).
Question 3
Explanation
Multiply the coefficients: \(7(-3)=-21\). The bases match, so add exponents: \(1+6=7\). The product is \(-21p^{7}\).
Question 4
Explanation
Multiply the coefficients: \(-4(-2)=8\). The bases match, so add exponents: \(3+2=5\). The product is \(8y^{5}\).
Question 5
Explanation
Multiply the coefficients: \(6(3)=18\). The bases match, so add exponents: \(5+4=9\). The product is \(18t^{9}\).
Question 6
Explanation
A power raised to a power multiplies exponents: \((x^{3})^{4}=x^{3\cdot4}=x^{12}\).
Question 7
Explanation
A power raised to a power multiplies exponents: \((b^{5})^{2}=b^{5\cdot2}=b^{10}\).
Question 8
Explanation
A power raised to a power multiplies exponents: \((q^{2})^{6}=q^{2\cdot6}=q^{12}\).
Question 9
Explanation
A power raised to a power multiplies exponents: \((r^{4})^{3}=r^{4\cdot3}=r^{12}\).
Question 10
Explanation
A power raised to a power multiplies exponents: \((w^{7})^{2}=w^{7\cdot2}=w^{14}\).
Question 11
Explanation
Cube every factor: \(2^3=8\), \((x^2)^3=x^6\), and \((y^1)^3=y^3\). Therefore the simplified form is \(8x^6y^3\).
Question 12
Explanation
Square the numerator and denominator: \(3^2=9\), \((a^2)^2=a^4\), and \((5b)^2=25b^2\). Therefore the simplified form is \(\frac{9a^4}{25b^2}\).
Question 13
Explanation
Subtract denominator exponent from numerator exponent: \(9-4=5\). Therefore the simplified form is \(z^5\).
Question 14
Explanation
Subtract exponents: \(m^{3-8}=m^{-5}\). Rewrite the negative power as \(\frac{1}{m^5}\). Therefore the simplified form is \(\frac{1}{m^5}\).
Question 15
Explanation
Cube every factor. The odd power preserves the negative sign, and \((q^2)^3=q^6\). Therefore the simplified form is \(-\frac{8p^3}{27q^6}\).
Question 16
Explanation
The complete nonzero base \(7u^3\) is raised to zero, so its value is \(1\). The correct choice is \(1\).
Question 17
Explanation
A negative exponent moves the powered base to the denominator: \(k^{-4}=\frac{1}{k^4}\). The correct choice is \(\frac{1}{k^4}\).
Question 18
Explanation
Because \(c^{-6}=\frac1{c^6}\), dividing by \(c^{-6}\) gives \(c^6\). The correct choice is \(c^6\).
Question 19
Explanation
Take the reciprocal, then square: \((\frac{5}{2a})^2=\frac{25}{4a^2}\). The correct choice is \(\frac{25}{4a^2}\).
Question 20
Explanation
The zero-exponent law follows from a same-base quotient, so the base must be nonzero. The correct choice is \(d\ne0\).
Question 21
Explanation
Divide coefficients and subtract exponents by base: \(18/6=3\), \(x^{7-2}=x^5\), and \(y^{3-5}=y^{-2}=1/y^2\). Thus the equivalent expression is \(\frac{3x^5}{y^2}\).
Question 22
Explanation
Multiply coefficients and add exponents for each base: \(a^{-2+5}=a^3\) and \(b^{4-1}=b^3\). Thus the equivalent expression is \(6a^3b^3\).
Question 23
Explanation
Square every factor: \(2^2=4\), \(m^{3\cdot2}=m^6\), and \(n^{-2\cdot2}=n^{-4}=1/n^4\). Thus the equivalent expression is \(\frac{4m^6}{n^4}\).
Question 24
Explanation
First multiply nested exponents to get \(p^8q^{12}\), then subtract denominator exponents: \(p^{8-5}q^{12-7}=p^3q^5\). Thus the equivalent expression is \(p^3q^5\).
Question 25
Explanation
The numerator is \(-\frac{s^3}{4r^2}\). Dividing by \(2r^{-1}=2/r\) multiplies by \(r/2\), giving \(-\frac{s^3}{8r}\). Thus the equivalent expression is \(-\frac{s^3}{8r}\).
Question 26
Explanation
Same-base multiplication gives \(a+7=19\), so \(a=12\). This value makes the exponents equal on both sides.
Question 27
Explanation
Power of a power gives \(4m=28\), so \(m=7\). This value makes the exponents equal on both sides.
Question 28
Explanation
The quotient law gives \(15-n=6\), hence \(n=9\). This value makes the exponents equal on both sides.
Question 29
Explanation
Multiply exponents: \(3(k-1)=18\). Then \(k-1=6\), so \(k=7\). This value makes the exponents equal on both sides.
Question 30
Explanation
Add exponents: \(2t+1+t-4=18\), so \(3t-3=18\) and \(t=7\). This value makes the exponents equal on both sides.
Question 31
Explanation
Move the decimal 5 places right to obtain \(7.3\). The original small nonzero decimal requires exponent \(-5\), so the normalized form is \(7.3\times10^{-5}\).
Question 32
Explanation
Move the decimal 6 places left to obtain \(4.82\). The original large value requires exponent \(6\), so the normalized form is \(4.82\times10^{6}\).
Question 33
Explanation
Move the decimal 3 places right to obtain \(9.16\). The original small nonzero decimal requires exponent \(-3\), so the normalized form is \(9.16\times10^{-3}\).
Question 34
Explanation
Move the decimal 4 places left to obtain \(7.35\). The original large value requires exponent \(4\), so the normalized form is \(7.35\times10^{4}\).
Question 35
Explanation
Move the decimal 4 places right to obtain \(-2.84\). The original small nonzero decimal requires exponent \(-4\), so the normalized form is \(-2.84\times10^{-4}\).
Question 36
Explanation
The exponent determines the decimal movement. Expanding \(3.7\times10^5\) gives \(370000\).
Question 37
Explanation
The exponent determines the decimal movement. Expanding \(8.04\times10^{-4}\) gives \(0.000804\).
Question 38
Explanation
The exponent determines the decimal movement. Expanding \(-6.2\times10^3\) gives \(-6200\).
Question 39
Explanation
The exponent determines the decimal movement. Expanding \(1.09\times10^{-6}\) gives \(0.00000109\).
Question 40
Explanation
The exponent determines the decimal movement. Expanding \(9.5\times10^7\) gives \(95000000\).
Question 41
Explanation
Multiply coefficients and add exponents: \(4.5(2)=9\) and \(6+(-3)=3\). The normalized result is \(9\times10^3\).
Question 42
Explanation
Divide coefficients and subtract exponents: \(7.2/2.4=3\) and \(8-3=5\). The normalized result is \(3\times10^5\).
Question 43
Explanation
Moving the decimal in 24 one place left increases the exponent by 1: \(24\times10^{-7}=2.4\times10^{-6}\). The normalized result is \(2.4\times10^{-6}\).
Question 44
Explanation
Coefficients give \(3(8)/6=4\). Exponents give \(-4+9-2=3\), so the result is \(4\times10^3\). The normalized result is \(4\times10^3\).
Question 45
Explanation
Move the coefficient decimal one place right from \(0.56\) to \(5.6\), so reduce the exponent by 1 to keep the value unchanged. The normalized result is \(5.6\times10^{11}\).
Question 46
Explanation
A power of a power multiplies exponents: \(4\cdot3=12\). The student incorrectly added them. Therefore \((x^4)^3=x^{12}\) is the only valid choice.
Question 47
Explanation
The negative sign belongs to the exponent and changes the factor's location across a fraction bar; it does not negate the value. Therefore \(A negative exponent indicates a reciprocal: v^{-3}=\frac1{v^3}\) is the only valid choice.
Question 48
Explanation
Changing 18 to 1.8 divides the coefficient by 10, so multiplying the power of ten by 10 raises the exponent to 6. Therefore \(1.8\times10^6\) is the only valid choice.
Question 49
Explanation
The product-of-powers law applies to multiplication. \(x^3+x^5\) is addition of unlike terms, so the exponents cannot be combined. Therefore \(x^3+x^5\) is the only valid choice.
Question 50
Explanation
The numerator is \(a^{-4}b^6\). Subtract denominator exponents: \(a^{-4-(-1)}b^{6-(-4)}=a^{-3}b^{10}=\frac{b^{10}}{a^3}\). Therefore \(\frac{b^{10}}{a^3}\) is the only valid choice.
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Questions to review
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- Question 1Product of powersEasy
- Question 2Product of powersEasy
- Question 3Product of powersEasy
- Question 4Product of powersEasy
- Question 5Product of powersEasy
- Question 6Power of a powerEasy
- Question 7Power of a powerEasy
- Question 8Power of a powerEasy
- Question 9Power of a powerEasy
- Question 10Power of a powerEasy
- Question 11Power of a productEasy
- Question 12Power of a quotientEasy
- Question 13Quotient of powersEasy
- Question 14Negative result exponentEasy
- Question 15Power of a quotientEasy
- Question 16Zero exponentMedium
- Question 17Negative exponentMedium
- Question 18Negative exponent in denominatorMedium
- Question 19Negative exponent on a fractionMedium
- Question 20Exponent-law conditionsMedium
- Question 21Multiple-variable quotientMedium
- Question 22Multiple-variable productMedium
- Question 23Power with negative exponentsMedium
- Question 24Combined exponent lawsMedium
- Question 25Multi-step exponent simplificationMedium
- Question 26Unknown exponentMedium
- Question 27Unknown exponentMedium
- Question 28Unknown exponentMedium
- Question 29Unknown exponentMedium
- Question 30Unknown exponentMedium
- Question 31Decimal to scientific notationMedium
- Question 32Decimal to scientific notationMedium
- Question 33Decimal to scientific notationMedium
- Question 34Decimal to scientific notationMedium
- Question 35Decimal to scientific notationMedium
- Question 36Scientific notation to decimalMedium
- Question 37Scientific notation to decimalMedium
- Question 38Scientific notation to decimalMedium
- Question 39Scientific notation to decimalMedium
- Question 40Scientific notation to decimalMedium
- Question 41Multiply scientific notationHard
- Question 42Divide scientific notationHard
- Question 43Normalize scientific notationHard
- Question 44Multi-step scientific notationHard
- Question 45Normalize scientific notationHard
- Question 46Exponent error analysisHard
- Question 47Negative exponent misconceptionHard
- Question 48Scientific notation error analysisHard
- Question 49Choose an exponent lawHard
- Question 50Advanced exponent simplificationHard