Operations on real numbers and the laws of exponents

The governing concept is rationalisation of a denominator containing a surd. Multiply the numerator and denominator by the conjugate of √10−3, namely √10+3. Then 1/(√10−3)=(√10+3)/[(√10−3)(√10+3)]. Using (a−b)(a+b)=a²−b², the denominator becomes 10−9=1. Hence the expression simplifies to √10+3. Therefore option A is correct. Option B is the original denominator expression and does not result from rationalisation. Option C incorrectly leaves a factor of 19 in the denominator, while option D expands the expression incorrectly and is not algebraically equivalent. The nonzero denominator is valid because √10 is not equal to 3.

When the base is same, exponents are added, so (2^3\cdot2^4=2^{3+4}=2^7). In exams, add exponents for same-base multiplication.

When the base is same, exponents are subtracted, so (\frac{5^6}{5^2}=5^{6-2}=5^4). For same-base division, subtract exponents.

In a power of a power, exponents are multiplied, so ((3^2)^4=3^{2\cdot4}=3^8). Adding the exponents here is a common mistake.

By the law of exponents, the zero power of any non-zero number is 1: \(a^0=1\), where \(a\ne0\). Therefore, 1 is correct. The options \(a\) and \(-a\) depend on the base, while 0 is not the general value of a zero power. Exam tip: the zero power of every non-zero number is 1.

A negative exponent moves the number to the denominator, so (10^{-2}=\frac{1}{10^2}=\frac{1}{100}). A negative exponent does not mean a negative value.

The power of a product applies to both factors, so ((2\cdot5)^3=2^3\cdot5^3). Apply the outside exponent to every factor inside the bracket.

The power of a fraction applies to both numerator and denominator, so \(\left(\frac{3}{4}\right)^2=\frac{3^2}{4^2}=\frac{9}{16}\). Do not square only the numerator.

When powers with the same base are multiplied, their exponents are added: \(7^2\cdot 7^3=7^{2+3}=7^5\). Therefore, option A is correct. Option B incorrectly gives the exponent as 6. Exam tip: in multiplication, keep the common base unchanged and add the exponents.

For division with the same base, exponents are subtracted, so (9^5\div9^3=9^{5-3}=9^2). Identify the law before calculating.

The governing exponent rule is that any nonzero number raised to the first power remains unchanged: a¹ = a. Applying this rule with a = 4 gives 4¹ = 4. Therefore option C is correct. Option B, 1, is the value of 4⁰, not 4¹. Option D, 16, is 4², so it uses the wrong exponent. Option A, 0, does not follow from any ordinary positive integral power of 4. The exponent tells how many times the base is used as a factor; with exponent 1, there is one factor of 4, not zero factors or two factors. Thus the expression has the simple value 4.

By the zero-exponent rule, the zeroth power of any non-zero number is 1, so \(6^0=1\). Also, \(2^3=8\). Therefore, \(6^0+2^3=1+8=9\), making option B correct. Exam tip: \(a^0=1\) applies when \(a\neq0\).

Using the order of operations, evaluate the powers first: \\(3^2=9\\) and \\(4^2=16\\). Their sum is \\(9+16=25\\), so the correct answer is 25. Option B, 49, is the square of 7, but the question asks for the sum of the squares of 3 and 4. Exam tip: calculate each exponent before performing the addition.

In division with the same base, exponents are subtracted, so (\frac{a^m}{a^n}=a^{m-n}). The condition (a\neq0) is important.

By the power-of-a-power law, the outer exponent \(n\) multiplies the inner exponent \(m\). Thus, \((a^m)^n=a^{mn}\), so option C is correct. Adding exponents, as in option A, applies when multiplying powers with the same base, not when raising one power to another. Exam tip: when a power is raised to another power, multiply the exponents.

The governing law is the power-of-a-product rule: (ab)ⁿ = aⁿbⁿ. The exponent 4 applies to the complete product xy, so (xy)⁴ means xy · xy · xy · xy. Regrouping the x factors and y factors gives x⁴y⁴. Therefore option A is correct. Option B raises only y to the fourth power and leaves x unchanged; option C does the reverse; and option D incorrectly changes multiplication into addition. It is important not to use the separate rule (a + b)ⁿ = aⁿ + bⁿ, which is generally false. Since x and y may be any suitable numbers, the symbolic product rule is the valid simplification.

The exponent of a fraction applies to both numerator and denominator. Hence \(\left(\frac{x}{y}\right)^3=\frac{x^3}{y^3}\).

\(2^5\) means multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\). Therefore, option B is correct. Option D represents adding 2 five times, which equals \(2 \times 5\), not \(2^5\). Exam tip: In \(a^n\), \(a\) is the base and \(n\) tells how many times the base is used as a factor.

Here the whole number ((-3)) is squared, so ((-3)^2=9). Parentheses include the negative sign in the power.

When exponents are the same, bases can be multiplied, so (2^3\cdot3^3=(2\cdot3)^3=6^3). This rule applies for equal exponents.