Operations on real numbers and the laws of exponents

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.

When bases are the same in multiplication, exponents are added, so (4^2\cdot4^3=4^5). Add exponents when bases are the same.

When bases are the same in division, exponents are subtracted, so (\frac{10^8}{10^5}=10^3). Subtract the lower exponent from the upper exponent.

Use the power-of-a-power law, (a^m)^n=a^(mn). Here the inner exponent is 4 and the outer exponent is 2, so (2^4)^2=2^(4×2)=2^8. Therefore option B is the standard simplified form. Option A, 2^6, comes from adding 4 and 2, but addition is not the rule for a power raised to a power. Option C, 2^16, results from multiplying the numerical powers themselves rather than multiplying the exponents. Option D, 4^4, is numerically equivalent because 4^4=(2^2)^4=2^8, but it is not the canonical form with the original base 2. Thus the intended answer is uniquely option B under standard simplification conventions.

By the laws of exponents, the zeroth power of every non-zero number is 1, so \(c^0=1\). This also follows from \(\frac{c^m}{c^m}=c^{m-m}=c^0\), since the quotient of a non-zero number by itself is 1. Option \(c\) represents \(c^1\), not \(c^0\). Exam tip: always check that the base is non-zero before applying the zero-exponent rule.

A negative exponent moves the base to the denominator, so (3^{-2}=\frac{1}{3^2}=\frac{1}{9}). A negative exponent does not mean a negative answer.

The law of the power of a product is \((ab)^n=a^n b^n\). Therefore, \((6\cdot7)^2=6^2\cdot7^2\), so option A is correct. In options B and C, the exponent is applied to only one factor, while option D represents a sum instead of a product. Exam tip: when a product is raised to a power, apply that power to every factor.

The power of a fraction applies to both numerator and denominator. Hence \(\left(\frac{7}{8}\right)^2=\frac{49}{64}\).

When powers with the same base are multiplied, their exponents are added: \(a^m \cdot a^n=a^{m+n}\). Therefore, \(12^2\cdot12^3=12^{2+3}=12^5\), so option A is correct. Option B incorrectly multiplies the exponents. Exam tip: for multiplication with the same base, add the exponents.

When powers with the same non-zero base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Thus, \(15^6 \div 15^2=15^{6-2}=15^4\), so option A is correct. Option B results from adding the exponents, which is the rule for multiplication, not division. Exam tip: subtract exponents for division and add them for multiplication when the bases are the same.

The first power of any non-zero number is the number itself, so \((21)^1=21\). The value 441 is \((21)^2\), not \((21)^1\). Exam tip: when the exponent is 1, the base remains unchanged.

The zeroth power of any non-zero number is 1, so \(4^0=1\). Also, \(5^2=25\). Therefore, \(4^0+5^2=1+25=26\), making option B correct. Exam tip: Remember that \(a^0=1\) for every non-zero number \(a\).

Evaluate the powers first: \(6^2=36\) and \(3^3=27\). Adding them gives \(36+27=63\), so the correct answer is 63. Values such as 54 or 45 can result from miscalculating a power or the final addition. Exam tip: evaluate exponents before carrying out addition or subtraction.

When powers with the same non-zero base are divided, their exponents are subtracted: \(\frac{w^a}{w^b}=w^{a-b}\). Hence, option B is correct. Option D reverses the order of subtraction, while option A represents the multiplication law and option C incorrectly multiplies the exponents. Exam tip: for division of powers with the same base, subtract the denominator exponent from the numerator exponent.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((s^5)^2=s^{5\times2}=s^{10}\), so option B is correct. Option A incorrectly adds the exponents instead of multiplying them. Exam tip: when a power is raised to another power, multiply the exponents.

By the power of a product rule, \((xy)^n=x^ny^n\). Therefore, \((ab)^5=a^5b^5\), so option A is correct. Option B incorrectly changes multiplication into addition, while options C and D apply the power to only one factor. Exam tip: when a product in parentheses is raised to a power, apply that power to every factor.

The exponent of a fraction applies to both numerator and denominator. Thus \(\left(\frac{m}{n}\right)^4=\frac{m^4}{n^4}\).

In \\(5^4\\), 5 is the base and 4 is the exponent. It means multiplying 5 by itself four times: \\(5\\cdot5\\cdot5\\cdot5\\)。 Therefore, option C is correct. Option B shows ordinary multiplication of 5 and 4, not the meaning of an exponent. Exam tip: Expand \\(a^n\\) as a product of n factors, each equal to \\(a\\).