Operations on real numbers and the laws of exponents

Using common denominator (6), (\frac{2}{3}=\frac{4}{6}). Hence (\frac{4}{6}+\frac{1}{6}=\frac{5}{6}).

Multiply and simplify: (\frac{3}{5}\cdot\frac{10}{9}=\frac{30}{45}=\frac{2}{3}). Cancelling first saves time.

According to the order of operations, multiplication is performed before subtraction. Thus, \\(3\\cdot2=6\\), and then \\(12-6=6\\). Option B results from incorrectly calculating \\(12-3\\) first. Exam tip: perform multiplication and division before addition and subtraction.

Evaluate the exponent first: \(3^2=9\). Then multiply to get \(9\cdot2=18\), and finally add \(2+18=20\). Therefore, option A is correct. Option D results from omitting the initial 2 in the final addition. Exam tip: follow the order exponents, multiplication/division, and then addition/subtraction.

4x^2 and 3x^2 are like terms because they have the same variable with the same exponent. Add only their coefficients: 4+3=7, while x^2 remains unchanged. Therefore, the simplified form is 7x^2. Option B is incorrect because exponents are not added when like terms are combined. Exam tip: For like terms, add or subtract the coefficients and keep the variable part unchanged.

The two terms are like terms because both contain x^3. Subtract their coefficients while keeping the common variable part unchanged: 6x^3 − 2x^3 = (6 − 2)x^3 = 4x^3. The exponent is not subtracted or added in this operation. Therefore option A is correct; 8x^3 would result from addition, while the other exponents use an invalid rule.

The expression \((2x+3y)\) has two terms, \(2x\) and \(3y\), separated by the plus sign. Parentheses do not change the number of terms. In an exam, count the parts separated by plus or minus signs.

The governing concept is the definition of like terms. Two algebraic terms are like terms when their variable parts are identical, including every variable and its exponent; only the numerical coefficients may differ. The term x² has x as its variable with exponent 2. In 3x², the variable part is also x², while the coefficient changes from the implied 1 to 3. Therefore 3x² is a like term and option A is correct. The term 3x has exponent 1, so it cannot be combined with x². The term x³ has the same letter but a different exponent. The term 2y² has the same exponent but a different variable. Thus likeness depends on the complete variable-and-exponent pattern, not merely on a shared letter, exponent, or numerical coefficient.

Multiplying the coefficients gives \(3\times2=6\). For the same base, add the exponents: \(x^2\cdot x^1=x^{2+1}=x^3\). Therefore, the product is \(6x^3\). Exam tip: multiply the numerical coefficients separately and add the exponents of identical variables.

Divide the numerical coefficients and subtract the exponents of the same base: \(18\div 6=3\) and \(a^5\div a^2=a^{5-2}=a^3\). Therefore, \(18a^5\div 6a^2=3a^3\), so option A is correct. Exam tip: when dividing powers with the same non-zero base, subtract the exponents; do not add them.

When powers with the same base are multiplied, their exponents are added: \(2^3\cdot2^0\cdot2^2=2^{3+0+2}=2^5\). Therefore, \(2^5\) is correct. The distractor \(2^6\) results from incorrectly treating the exponent 0 as 1. Exam tip: for every non-zero number, \(a^0=1\).

For powers with the same nonzero base, multiplication adds exponents and division subtracts exponents: aᵐ·aⁿ=aᵐ⁺ⁿ and aᵐ÷aⁿ=aᵐ⁻ⁿ. First simplify the numerator: 5²·5³=5⁵. Then divide by 5⁴: 5⁵÷5⁴=5⁵⁻⁴=5¹. Hence option A is correct, and the numerical value is 5. Option B results from stopping after multiplying and forgetting the division. Option C incorrectly adds all three exponents, even though the final operation is division. Option D subtracts in the reverse order, 4−5, rather than numerator exponent minus denominator exponent. Since the common base 5 is nonzero, the exponent laws apply without any restriction problem.

Here \(\left(\frac{2}{3}\right)^0=1\) and \(\left(\frac{1}{2}\right)^2=\frac{1}{4}\). Therefore the sum is \(\frac{5}{4}\).

When the base is the same, exponents are added, so (3^2\cdot3^5=3^7). Add exponents when bases are equal.

When the base is the same, exponents are subtracted, so (\frac{8^7}{8^4}=8^3). Subtract exponents for same-base division.

The governing rule is the power-of-a-power law: (aᵐ)ⁿ=aᵐⁿ. The outer exponent multiplies the inner exponent, so (4³)²=4^(3×2)=4⁶. Therefore option B is correct. The exponents are multiplied, not added; consequently 4⁵ is not obtained from this expression. The value 4⁹ would result from another incorrect combination of the exponents. Option D, 4³, ignores the outer square and therefore also fails to preserve the value. The same result can be checked numerically: 4³=64 and 64²=4096, while 4⁶=4096. The revised fourth option is deliberately 4³ rather than an equivalent expression such as 16³, ensuring that only one listed option represents the intended simplified form.

By the laws of exponents, the zero power of every non-zero number is 1: \(b^0=1\), provided \(b\neq0\). The option \(b\) is incorrect because it represents \(b^1\), not \(b^0\). Exam tip: whenever the base is non-zero, its zero power is always 1.

A negative exponent moves the base to the denominator, so (2^{-3}=\frac{1}{2^3}=\frac{1}{8}). A negative exponent does not mean a negative answer.

The power of a product applies to every factor, so ((3\cdot4)^2=3^2\cdot4^2). Apply the exponent to all factors inside the bracket.

The power of a fraction applies to both numerator and denominator. Therefore \(\left(\frac{5}{6}\right)^2=\frac{25}{36}\).