Example Question - negative exponents
Here are examples of questions we've helped users solve.
Simplification of an Algebraic Fraction with Negative Exponents
\[ \left( \frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3} \right)^2 = \left( \frac{4a^6b^5c^{-2}}{8a^{-12}b^{18}c^3} \right)^2 = \left( \frac{4 \cdot a^{6-(-12)} \cdot b^{5-18} \cdot c^{-2-3}}{8} \right)^2 = \left( \frac{4 \cdot a^{18} \cdot b^{-13} \cdot c^{-5}}{8} \right)^2 = \left( \frac{a^{18} \cdot b^{-13} \cdot c^{-5}}{2} \right)^2 = \frac{a^{36} \cdot b^{-26} \cdot c^{-10}}{4} \]
Solving Exponential Expression
Por supuesto, vamos a resolver la expresión paso a paso. Tenemos la siguiente expresión: \((-10)^4 \cdot (-10) \cdot (-10)^5 \div (-10)^3\) Lo primero que debemos recordar es la propiedad de las potencias que indica que cuando multiplicamos potencias de la misma base, sumamos los exponentes. Además, cuando dividimos potencias de la misma base, restamos los exponentes. Comenzamos con la parte de la multiplicación de potencias: \((-10)^4 \cdot (-10)^1 \cdot (-10)^5 = (-10)^{4+1+5} = (-10)^{10}\) Ahora, dividimos entre \((-10)^3\): \(\frac{(-10)^{10}}{(-10)^3} = (-10)^{10-3} = (-10)^7\) El resultado de \((-10)^7\) es un número negativo porque cualquier número negativo elevado a un exponente impar es negativo, y su valor absoluto es \(10^7\) porque \(|-10|^7 = 10^7\). En resumen: \((-10)^4 \cdot (-10) \cdot (-10)^5 \div (-10)^3 = (-10)^7\) Y el valor de \((-10)^7\) es \(-10,000,000\), que en español sería diez millones negativos.
Solving Exponential Expression with Division
La expresión que aparece en la imagen es: \[ C = [(-10)^4 \cdot (-10) \cdot (-10)^{35}] : [(-10)^{33}] \] Para resolver esta expresión, primero simplificamos y usamos las propiedades de los exponentes. Comencemos con el numerador de la expresión. Cuando multiplicamos números con la misma base, sumamos los exponentes de la base. Luego, se maneja el signo negativo usando la propiedad de que un número negativo elevado a una potencia par es positivo y a una potencia impar es negativo. \[ (-10)^4 \cdot (-10) \cdot (-10)^{35} = 10^4 \cdot -10^1 \cdot (-10)^{35} \] \[ = 10^4 \cdot -10^{1+35} \] \[ = 10^4 \cdot -10^{36} \] \[ = -10^{4+36} \] \[ = -10^{40} \] Ahora, simplificamos el denominador: \[ (-10)^{33} \] Un número negativo elevado a una potencia impar sigue siendo negativo: \[ = -10^{33} \] Ahora dividimos el numerador por el denominador: \[ \frac{-10^{40}}{-10^{33}} \] Cuando dividimos números con la misma base, restamos los exponentes: \[ = 10^{40-33} \] \[ = 10^7 \] Por lo tanto, la respuesta final es \( 10^7 \).
Simplifying Exponential Expression
The expression given in the image is: \[ \left( \frac{8y^4z^8}{16y^8z} \right)^4 \] To simplify this expression, first, simplify the fraction by canceling common factors and then apply the exponent of 4: \[ \left( \frac{8}{16} \cdot \frac{y^4}{y^8} \cdot \frac{z^8}{z} \right)^4 \] Simplify the fractions: \[ \left( \frac{1}{2} \cdot y^{4-8} \cdot z^{8-1} \right)^4 \] which simplifies further to: \[ \left( \frac{1}{2} \cdot y^{-4} \cdot z^7 \right)^4 \] Now apply the exponent of 4 to each term within the parentheses: \[ \left( \frac{1}{2} \right)^4 \cdot y^{-4 \cdot 4} \cdot z^{7 \cdot 4} \] This gives: \[ \frac{1}{16} \cdot y^{-16} \cdot z^{28} \] Since y has a negative exponent, it can be moved to the denominator: \[ \frac{z^{28}}{16y^{16}} \] This is the simplified form of the original expression.
Solving a Negative Fractional Exponent Expression
The expression in the image is \((\frac{8}{27})^{-2/3}\). To solve this, we can apply the rule for negative exponents and fractional exponents. A negative exponent means that you take the reciprocal of the base, and a fractional exponent means you take the root of the base (the denominator of the fraction) and then raise it to the power of the numerator. Here's the step-by-step calculation: \[ \left(\frac{8}{27}\right)^{-2/3} = \left(\frac{27}{8}\right)^{2/3} \] Now we take the cube root of both the numerator and the denominator and then square them because the exponent is \(2/3\): \[ \left(\sqrt[3]{\frac{27}{8}}\right)^2 = \left(\frac{\sqrt[3]{27}}{\sqrt[3]{8}}\right)^2 = \left(\frac{3}{2}\right)^2 \] Finally, we square the result: \[ \left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4} \] Therefore, \((\frac{8}{27})^{-2/3}\) equals \(\frac{9}{4}\) or 2.25.
Simplifying Algebraic Expression with Exponents
The expression in the image is \[ x^2 \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \div x^2 \] First, we'll simplify the term in the denominator of the first fraction by moving it to the numerator with a negative exponent: \[ x^2 \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \] Now, we'll combine the terms by finding a common exponent for \( x \). In this case, we want to match the exponent of \( x \) to the smallest exponent, which is \( -2 \): \[ x^{2-2} \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \] Upon simplifying \( x^{2-2} \), we get: \[ (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \] The expression cannot be simplified further because the terms are not like terms (they have different exponents). Therefore, the simplified expression is: \[ (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \]
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