Operations on real numbers and the laws of exponents

The zero-exponent rule states that the zero power of every non-zero number is 1. Therefore, \(u^0=1\), \(v^0=1\), and \(2^0=1\). Hence, \(u^0+v^0+2^0=1+1+1=3\), so option C is correct. Option B may result from counting only two terms, but all three terms are equal to 1. Exam tip: apply \(a^0=1\) only when \(a\ne0\).

The negative-exponent law is \(x^{-n}=\frac{1}{x^n}\), where \(x\ne0\). Therefore, \(x^{-4}=\frac{1}{x^4}\), so option B is correct. Option A incorrectly drops the negative sign without taking the reciprocal, while option D treats the exponent 4 as a factor. Exam tip: For a negative exponent, write the base in the denominator and change the exponent to positive.

Using the identity \((a-b)^2=a^2-2ab+b^2\), with \(a=x\) and \(b=y\), we get \((x-y)^2=x^2-2xy+y^2\). Hence, option B is correct. Option C has a positive middle term and represents the expansion of \((x+y)^2\). In exams, remember that the middle term of \((a-b)^2\) is \(-2ab\).

Using the distributive law, multiply the outside factor 3 by each term inside the bracket: \(3(x+4)=3\times x+3\times4=3x+12\). Therefore, option B is correct. Option C incorrectly omits the factor 3 from the first term. Exam tip: multiply the number outside the bracket by every term inside it.

8a and 6a are like terms because both contain the variable a to the first power. Add their coefficients: 8a+6a=(8+6)a=14a. The exponent of a does not change, so 14a^2 is incorrect. Exam tip: Add or subtract coefficients only when the variable part is identical.

When powers have the same base, their exponents are added, so \(a^m \times a^n=a^{m+n}\). The form \(a^{m-n}\) belongs to division of like bases. Exam tip: check that the bases are identical before applying this rule.

The product of the coefficients is \(5\cdot4=20\). For powers with the same base, add the exponents: \(x^1\cdot x^3=x^{1+3}=x^4\). Hence, the product is \(20x^4\). Exam tip: multiply the numerical coefficients separately and add exponents of the same variable.

Divide the numerical coefficients and subtract exponents of the same nonzero base: 24 ÷ 6 = 4 and x^5 ÷ x^2 = x^(5−2) = x^3. Hence (24x^5)/(6x^2) = 4x^3. The condition x ≠ 0 ensures that division by x^2 is defined. Option B subtracts coefficients incorrectly, while C adds exponents and D multiplies unrelated factors.

The product rule states \(a^m \times a^n=a^{m+n}\) for the same base, so exponents are added during multiplication. In the quotient rule, exponents are subtracted. Exam tip: first check that the bases are identical.

First evaluate the two powers: \\(3^2=9\\) and \\(3^3=27\\). Therefore, \\(3^2+3^3=9+27=36\\), so the correct answer is 36. Remember that exponents are not added in addition; \\(3^2+3^3\\) cannot be written as \\(3^5\\). Exam tip: for powers with the same base, exponents are added only during multiplication, not addition.

The distributive law states that multiplying a number by a sum is the same as multiplying it separately by each term: \\(m(n+p)=mn+mp\\). The commutative and associative laws deal with changing order and grouping, respectively, so they do not describe this expansion. Exam tip: when a factor outside parentheses is multiplied by every term inside, identify the distributive law.

In this equation, interchanging x and y does not change the sum: x+y=y+x. Therefore, it represents the commutative law of addition. The distributive law involves multiplying over a sum, such as x(y+z)=xy+xz. In an exam, if changing the order of terms leaves the result unchanged, identify the commutative law.

In this equation, the order of the terms remains the same, but their grouping changes. The sum remains unchanged when the grouping is changed, which is the associative law. It is not the commutative law, because that law changes the order of the terms, as in \(x+y=y+x\). Exam tip: associative means changing grouping, while commutative means changing order.

When adding decimals, align the decimal points: \(0.75+0.15=0.90\). Therefore, the correct answer is 0.90. The value 0.80 is too small, while 1.00 is a close but incorrect estimate. In exams, check the alignment of decimal points before adding.

Using common denominator (8), (\frac{3}{4}=\frac{6}{8}). Hence (\frac{6}{8}+\frac{1}{8}=\frac{7}{8}).

Multiply and simplify: (\frac{4}{7}\cdot\frac{14}{5}=\frac{56}{35}=\frac{8}{5}). Cancelling first makes calculation easier.

According to the order of operations, multiply first: \(4\cdot3=12\). Then subtract: \(15-12=3\). The answer 33 would result from subtracting first and then multiplying by 3, which is not the correct order. Exam tip: multiplication and division are performed before addition and subtraction.

Exponents and multiplication are performed before addition. Thus, \\(2^3=8\\), then \\(8\cdot4=32\\), and finally \\(5+32=37\\). Therefore, the correct answer is 37. In an exam, remember the BODMAS order: exponentiation, multiplication, and then addition; 52 results from using the operations in the wrong order.

\(9x^2\) and \(5x^2\) are like terms because they have the same variable raised to the same power. Therefore, add their coefficients: \((9+5)x^2=14x^2\). The exponent does not change during addition, so \(14x^4\) is incorrect. Exam tip: when adding like terms, add only the coefficients and keep the variable part unchanged.

Both terms are like terms because they have the same variable x raised to the fourth power. Subtract only the coefficients: 11x^4 − 7x^4 = (11 − 7)x^4 = 4x^4. The common factor x^4 remains unchanged. Option B incorrectly adds the coefficients, option C changes the exponent to zero without division, and option D incorrectly adds exponents.

The expression \(4x+5y+2\) contains three separate terms: \(4x\), \(5y\), and \(2\). Terms are separated by plus or minus signs, so the correct answer is 3. Choosing 2 by counting only the variable terms is incorrect because the constant \(2\) is also a term. Exam tip: every constant, including a positive or negative one, is counted as a term.

To identify a like term, compare the variable part carefully. The variable itself and every exponent must be identical; the coefficient may change. The reference term \(y^3\) contains the variable \(y\) raised to the third power. Therefore a like term must also have exactly \(y^3\), even if a different number is placed before it.

Option A is \(7y^3\), so it has the required variable and exponent and differs only in coefficient. Option B has exponent 2, option C has exponent 4, and option D uses \(x\) instead of \(y\). None of those can be combined with \(y^3\) as like terms. Hence option A is the correct choice.

When powers with the same base are multiplied, their exponents are added: \(x^3 \cdot x^2=x^{3+2}=x^5\). The coefficients multiply to \(4\cdot3=12\). Therefore, \(4x^3\cdot3x^2=12x^5\). Exam tip: multiply the coefficients and add the exponents of the same variable.

Divide the numerical coefficients and use the quotient law for powers: \(\frac{30}{5}=6\) and \(\frac{a^6}{a^3}=a^{6-3}=a^3\). Thus, \(\frac{30a^6}{5a^3}=6a^3\). Option B adds the exponents instead of subtracting them, while options C and D use incorrect coefficient operations. Exam tip: when dividing powers with the same non-zero base, subtract the exponents.

When powers with the same base are multiplied, their exponents are added: \(4^2\cdot4^0\cdot4^3=4^{2+0+3}=4^5\). Since \(4^0=1\), the zero exponent does not change the result. Exam tip: add exponents when multiplying powers with the same base.